# Rational curvature invariants for 2-complexes

Henry Wilton

###### Abstract

New invariants for 2-dimensional cell complexes are defined, which can be
interpreted as curvature bounds. These invariants are proved to be rational
and computable in a companion article. This document is a survey that collects
theorems about these invariants, computes examples, and lays out a programme
of conjectures.

###### Contents

  1. 1 Introduction
  2. 2 Branching and area
  3. 3 Group pairs
  4. 4 Irreducible complexes
  5. 5 Surfaces
  6. 6 Curvature invariants
  7. 7 Inequalities and first examples
  8. 8 Nielsen equivalence
  9. 9 Irrigid complexes
  10. 10 Relations to other invariants
    1. 10.1 Deficiency
    2. 10.2 Primitivity rank
    3. 10.3 Stable commutator length
  11. 11 Examples
    1. 11.1 Angle structures and $CAT(\kappa)$ spaces
    2. 11.2 Small-cancellation and random complexes
    3. 11.3 One-relator groups
    4. 11.4 Geometric 2-complexes
    5. 11.5 Graphs of free groups
    6. 11.6 A botanical map
  12. 12 Properties of groups with curvature bounds
    1. 12.1 Concise 2-complexes
    2. 12.2 Non-positive irreducible curvature
    3. 12.3 Negative irreducible curvature
    4. 12.4 Surface curvature bounds

##  1 Introduction

Combinatorial group theory can be characterised as the study of groups $G$
defined by (usually finite) presentations. One might equivalently define the
subject as the study of the fundamental groups of (usually compact)
2-dimensional cell complexes $X$. One of the central insights that arose from
the advent of geometric group theory in the 1980s was that imposing curvature
bounds on $X$ or $G$ can lead to powerful theorems, such as solutions to
algorithmic problems, bounds on the cohomological dimension, or information
about the structure of possible subgroups or quotients.

Our purpose here is to associate new curvature invariants to any finite
2-complex $X$, starting with the following crude definition. The _average
curvature_ of a finite 2-complex $X$ is defined to be

$\kappa(X):=\frac{\chi(X)}{\mathrm{Area}(X)}$

where $\chi(X)$ is the Euler characteristic of $X$ and $\mathrm{Area}(X)$ is
the number of 2-cells of $X$. This definition is motivated by the Gauss–Bonnet
theorem, which tells us that the average Gaussian curvature of a closed
Riemannian surface $S$ is $2\pi\chi(S)/\mathrm{Area}(S)$.

The average curvature is not very different from the Euler characteristic, and
so will not on its own contain much information about $X$ or its fundamental
group. However, it is a familiar idea in topology and geometry that we may
define interesting invariants of a space $X$ by probing it with maps from
other spaces $Y$. Following this idea, we define more refined invariants by
extremising $\kappa$ over certain classes of 2-complexes $Y$ that map to $X$.
(In fact, it turns out to be fruitful to allow $Y$ to be a slightly more
general object called a _branched_ 2-complex.111See §2 for the definition of
branched 2-complexes and §6 for the definition of $\kappa(X)$ in that
context.)

There are various ways in which it may be possible to simplify a 2-complex
$X$, such as if $X$ has a free face, a separating vertex etc. A 2-complex that
cannot be simplified in any of these ways is called _irreducible_. Our first
curvature invariant is the _maximal irreducible curvature_

$\rho_{+}(X):=\sup_{Y\in\mathrm{Irred}(X)}\kappa(Y)$

where $\mathrm{Irred}(X)$ consists of all finite, irreducible, branched
2-complexes $Y$ admitting essential maps $Y\to X$. Likewise, the _minimal
irreducible curvature_ $\rho_{-}(X)$ is the corresponding infimum.222See §2
for the definition of essential maps, §4 for the definition of irreducible
complexes, and §6 for the precise definition of $\rho_{\pm}(X)$.

Although we believe that these definitions have not appeared in the literature
before, they are prefigured in various places. In the early 2000s, Wise
defined the _non-positive-_ and _negative-immersions_ properties for
2-complexes, as a route towards studying the fundamental group $\pi_{1}(X)$,
especially in the case where $X$ is a one-relator presentation complex [50,
51]. These properties are in a similar spirit to the definitions of non-
positive and negative irreducible curvature, respectively, and indeed this
document can be thought of as a modification of the programme suggested by
Wise in [52] and [53]. The independent proofs that one-relator groups have
non-positive immersions by Helfer and Wise [21] and by Louder and the author
[27], and the subsequent work on various notions of negative immersions for
one-relator groups [28, 29, 30, 53], provide large classes of examples to
which these definitions apply. Puder’s _primitivity rank_ [40] is another
closely related invariant, discussed in §10.2.

Irreducible 2-complexes are the largest naturally defined class of finite
2-complexes over which we can usefully extremise $\kappa$. We can define a
second pair of invariants by extremising $\kappa$ over the smallest naturally
defined class of irreducible 2-complexes mapping to $X$, namely closed
surfaces. The _maximal surface curvature_ of $X$ is defined to be

$\sigma_{+}(X):=\sup_{Y\in\mathrm{Surf}(X)}\kappa(Y)$

where $\mathrm{Surf}(X)$ consists of all closed surfaces $Y$ admitting
essential maps $Y\to X$, and the _minimal surface curvature_ ,
$\sigma_{-}(X)$, is defined to be the corresponding infimum.333See §6 for the
definitions $\sigma_{\pm}(X)$. In contrast to the case of irreducible
curvature, it is not immediately obvious that $\mathrm{Surf}(X)$ is non-empty
for most complexes $X$. However, it follows from the results of [48] that
$\mathrm{Surf}(X)\neq\varnothing$ whenever $X$ is irreducible, so the
invariants $\sigma_{\pm}(X)$ are well defined in that case. (See Theorem 5.2.)

Again, although we believe these definitions to be new, they do have various
precedents in the literature. Wise’s _planar sectional curvature_ [51]
provides sufficient conditions for $X$ to have negative or non-positive
surface curvature. In a different direction, the quantity
$(1-\sigma_{+}(X))/2$ can be regarded as an _unoriented_ version of stable
commutator length; see §10.3 for a discussion. The two most important theorems
about stable commutator length in free groups – the Duncan–Howie lower bound
$\mathrm{scl}(w)\geq 1/2$ [17] and Calegari’s rationality theorem [10] –
prefigure some of the most important theorems that we discuss here.

The fundamental result about our invariants is the _rationality theorem_ ,
which tells us that the extrema in the above definitions are realised, and in
so doing justifies the words ‘maximal’ and ‘minimal’ in the names of the
curvatures. The proof realises $\rho_{\pm}(X)$ and $\sigma_{\pm}(X)$ as
extrema of rational linear programming problems, along the lines of Calegari’s
rationality theorem for stable commutator length in free groups [10]. In
particular, as well as being rational, these invariants can be computed, at
least in principle.

######  Theorem 1.1 (Rationality theorem).

If $X$ is a finite, irreducible 2-complex, then

$\rho_{+}(X)=\max_{Y\in\mathrm{Irred}(X)}\kappa(Y)$

and

$\sigma_{+}(X)=\max_{Y\in\mathrm{Surf}(X)}\kappa(Y)\,.$

Likewise, $\rho_{-}(X)$ and $\sigma_{-}(X)$ are the corresponding minima. In
particular, $\rho_{\pm}(X),\sigma_{\pm}(X)\in\mathbb{Q}$. Furthermore, they
can be computed algorithmically from $X$.

The proof of Theorem 1.1 is long and technical, and is therefore in a self-
contained paper [49].

Theorem 1.1 provokes the question of which ranges of values the curvature
invariants can take. The next theorem constrains them inside a certain three-
dimensional simplex.

######  Theorem A (Curvature inequalities).

If $X$ is a finite, irreducible 2-complex, then

$-\infty<\rho_{-}(X)=\sigma_{-}(X)\leq\sigma_{+}(X)\leq\rho_{+}(X)\leq 1\,.$

Of the various relationships contained in Theorem A, much the most striking
and difficult to prove is that $\rho_{-}(X)=\sigma_{-}(X)$, which means that
the minimal irreducible curvature is always realised by an essential map from
a branched surface.

These results motivate a broad programme, which can be summarised in two
problems that we call, for ease of reference, the _botany problem_ and the
_geography problem_. Given the historical emphasis in geometric group theory
on upper curvature bounds, the cases of _non-positive curvature_ (i.e.
$\rho_{+}(X)\leq 0$ or $\sigma_{+}(X)\leq 0$) and _negative curvature_ (i.e.
$\rho_{+}(X)<0$ or $\sigma_{+}(X)<0$) are of particular interest.

The botany problem asks for examples of 2-complexes satisfying curvature
bounds.

######  Problem 1.2 (Botany problem).

For natural classes of 2-complexes $X$, compute the invariants $\rho_{+}(X)$,
$\sigma_{+}(X)$. Find examples of 2-complexes $X$ that are _non-positively
curved_ or _negatively curved_.

In this paper, we collect several solutions to the botany problem. Some of the
most significant are summarised in the next theorem.

######  Theorem B (Examples).

Let $X$ be a finite, irreducible 2-complex.

  1. (i)

If $X$ is locally CAT(0) or satisfies the C(6) small-cancellation condition
then $\sigma_{+}(X)\leq 0$. If $X$ is locally CAT(-1) or C(7) then
$\sigma_{+}(X)<0$.

  2. (ii)

If $X$ is the presentation complex of a torsion-free one-relator group then
$\rho_{+}(X)\leq 0$.

  3. (iii)

If $X$ is is the spine of a compact 3-manifold444Formally, this hypothesis
asks that $X$ should be _geometric_ in the sense of Definition 11.22. then
$\sigma_{+}(X)=\rho_{+}(X)$.

Item (i) of the theorem is the content of Corollaries 11.6 and 11.7 together
with Propositions 11.10 and 11.11, item (ii) is Theorem 11.15, and item (iii)
is Corollary 11.24.

These results should be compared with the standard notions of curvature in
geometric group theory. Item (i) of Theorem B suggests that $\sigma_{+}(X)$
carries similar information to the CAT($\kappa$) and small-cancellation
notions of curvature, which are already widely used throughout geometric group
theory. On the other hand, item (ii) implies that our notions of non-positive
curvature encompass some examples, such as the presentation complexes of
Baumslag–Solitar groups, that do not satisfy any of the standard notions of
non-positive curvature in geometric group theory. Finally, item (iii) suggests
that irreducible and surface curvatures capture information that explains some
of the special properties of 3-manifold groups.

These examples are discussed in §11, along with some conjectures which posit
that the scope of our invariants can be expanded to capture the behaviour of
many of the other favourite low-dimensional examples of geometric group
theory. Figure 1 illustrates examples of 2-complexes $X$ with various
combinations of $\rho_{+}(X)$ and $\sigma_{+}(X)$.

Our second broad problem is the geography problem, which asks for constraints
on 2-complexes that satisfy curvature bounds.

######  Problem 1.3 (Geography problem).

Prove theorems about 2-complexes $X$ satisfying constraints on $\rho_{\pm}(X)$
or $\sigma_{\pm}(X)$, especially under the hypotheses that either
$\rho_{+}(X)$ or $\sigma_{+}(X)$ is negative or non-positive.

We offer a few contributions to the geography problem here. The first is an
easy observation, which characterises _concise_ 2-complexes.

######  Definition 1.4.

A 2-complex is called _concise_ unless it has either two 2-cells with
homotopic attaching maps or a single 2-cell whose boundary is a proper power.

Equivalently, for a group presentation $\langle a_{1},\ldots,a_{m}\mid
r_{1},\ldots,r_{n}\rangle$, the associated presentation complex $X$ is concise
if the family of cyclic subgroups $\\{\langle r_{1}\rangle,\ldots,\langle
r_{n}\rangle\\}$ of the free group $F_{m}=\langle a_{1},\ldots,a_{m}\rangle$
is malnormal. This definition is slightly stronger than the definition of a
concise 2-complex used by Chiswell–Collins–Huebschmann [14], but is convenient
for our purposes.

######  Proposition C.

Let $X$ be a finite, irreducible 2-complex. If $\rho_{+}(X)=1$ then
$\sigma_{+}(X)=1$, and $X$ is not concise.

The proof is given in §12.1. The proposition implies that there is no
2-complex $X$ with $\sigma_{+}(X)<\rho_{+}(X)=1$. I expect this to be the only
collection of curvature constraints that are not realised by any finite,
irreducible 2-complex.

The next contribution is more interesting, and illustrates the strength of
these invariants. It is not difficult to see that any closed surface $X$ has
constant irreducible curvature – that is, $\rho_{+}(X)=\rho_{-}(X)$ (see
Example 7.5). Remarkably, 2-complexes with constant irreducible curvature can
be completely classified, and they are all closely related to surfaces. More
precisely, they are _irrigid_ , which roughly means that they can be obtained
from surfaces with boundary by gluing their boundary circles together.555See
Definition 9.1 for a precise definition.

######  Theorem D.

Let $X$ be a finite, irreducible 2-complex. If $\rho_{+}(X)=\rho_{-}(X)$ then
$X$ is irrigid.

It is doubtful that any other collection of curvature constraints leads to a
complete characterisation beyond Proposition C and Theorem D. However,
curvature constraints can have profound consequences for the group theory of
the fundamental group. The next theorem collects some examples of such
consequences, which all derive from the work of Wise, Louder and the author.
Recall that a group is _locally indicable_ if every non-trivial finitely
generated subgroup surjects $\mathbb{Z}$; it is _2-free_ if every 2-generator
subgroup is free; and it is _coherent_ if every finitely generated subgroup is
finitely presented.

######  Theorem E (Upper bounds on irreducible curvature).

Let $X$ be a finite, irreducible 2-complex.

  1. (i)

If $\rho_{+}(X)\leq 0$ then $X$ is aspherical and $\pi_{1}(X)$ is locally
indicable.

  2. (ii)

If $\rho_{+}(X)<0$ then $\pi_{1}(X)$ is 2-free and coherent.

Item (i) of Theorem E is Theorem 12.3, while item (ii) is a sample of the
conclusions of Theorem 12.8.

Conjecturally, upper bounds on irreducible curvature should have even stronger
consequences; see, for instance, Conjectures 12.4 and 12.9. There are also
corresponding conjectures about 2-complexes with upper bounds on their surface
curvature; see Conjectures 12.10, 12.11 and 12.12. Figure 2 gives a
conjectural picture of the properties of $X$ that are determined by
$\rho_{+}(X)$ and $\sigma_{+}(X)$.

This paper is structured as follows. The basic definitions of branched
2-complexes and the maps between them are given in §2. The fundamental notion
of an irreducible complex is introduced in §4, and surfaces appear in §5. The
stage is then set to define $\rho_{\pm}(X)$ and $\sigma_{\pm}(X)$ in §6. Some
immediate consequences of the definitions are explained in §7. One advantage
of these definitions over other similar curvature notions is that they are
_Nielsen invariants_ , meaning that they do not change if the 1-skeleton is
modified by a homotopy equivalence; this is explained in §8. Theorems A and D
are proved in §9, where irrigid complexes are discussed. Relationships with
other similar invariants are discussed in §10. Finally, §11 addresses the
botany problem by collecting examples, and §12 addresses the geography problem
by collecting properties of 2-complexes with curvature bounds.

### Acknowledgements

This work would have been impossible without the input of Lars Louder. Our
collaborative work on one-relator groups convinced me that these definitions
were sufficiently broadly applicable to be worth studying in their own right.
The debt that this work also owes to the vision of Dani Wise cannot be
overstated – it’s clear from reading some of his early papers on coherence
[50, 51] that he anticipated much of what is written here. Danny Calegari’s
ideas, and especially to his proof of the rationality theorem for stable
commutator length in free groups [10], have also had a profound influence.

Thanks are also due to Edgar Bering, Martin Blufstein, Daniel Groves, Nir
Lazarovich, Jason Manning and Doron Puder for useful conversations.

##  2 Branching and area

A 2-dimensional complex $X$ is the result of attaching 2-cells along to a
topological graph along their boundaries. This graph is the 1-skeleton
$X_{(1)}$, and is constructed by attaching gluing edges from an edge set
$E_{X}$ to a vertex set $V_{X}$.666Several formalisms for graphs can be used.
For instance, $X_{(1)}$ can be taken to be a Serre graph, as in Stallings’
definitive paper [44]. Our discussion here does not rely on the details of the
formalism chosen. The 2-cells of a 2-complex $X$ are called _faces_ , and the
set of faces is denoted by $F_{X}$. Formally, the faces can be modelled as
circles mapping into $X_{(1)}$. After a homotopy, we may always assume that
the attaching maps of the faces are immersions, at the cost of removing any
face whose attaching maps is homotopically trivial in the 1-skeleton.

######  Example 2.1 (Presentation complex).

A group presentation $\mathcal{P}=\langle A\mid R\rangle$ defines a
presentation complex $X_{\mathcal{P}}$, as long as every relator $r\in R$ is
non-trivial.

The maps we allow between 2-complexes are _combinatorial_ , meaning that the
send vertices to vertices, edges to edges and faces to faces, all
homeomorphically.

This much is standard, but it turns out to be fruitful to extend the category
of 2-complexes that we work with in a slightly non-standard way.

######  Definition 2.2.

A _branched 2-complex_ $X$ is a 2-complex equipped with a notion of _area_ ,
namely a map

$\mathrm{Area}:F_{X}\to\mathbb{Z}_{>0}\,.$

The notation $\mathrm{Area}(X)$ is also used to denote the sum of the areas of
the faces of $X$, and we insist that $\mathrm{Area}(X)>0$.

We also allow a slightly wider selection of maps between branched 2-complexes.
A _branched morphism_ between branched 2-complexes $\phi:Y\to X$ still sends
vertices to vertices and edges to edges homeomorphically. It also send faces
to faces, but on faces the maps are allowed a single ramification point in the
centre; that is, on the faces, a map is modelled on the polynomial map

$z\mapsto z^{m}$

of the unit disc in the complex plane, for some positive integer $m$. In
particular, $\phi$ assigns to each face $f$ of $Y$ a _multiplicity_
$m_{\phi}(f)\in\mathbb{N}$, namely the ramification index at the centre of
$f$. However, the branching of the map $\phi$ is constrained to respect area
in the following sense:

$\mathrm{Area}(f)=m_{\phi}(f)\mathrm{Area}(\phi(f))$

for each face $f\in F_{Y}$.

Each 2-complex has a canonical structure as a branched 2-complex, obtained by
setting the area of each face equal to 1. We will sometimes call such a
2-complex _standard_ , to emphasise that there is no branching. Note that a
branched morphism between two standard 2-complexes is just a combinatorial map
in the standard sense.

Branched coverings of surfaces provide a wealth of natural examples, as long
as one is careful to place the branch points in the centres of faces.

######  Example 2.3 (Branched self-coverings of the 2-sphere).

For each $n\geq 1$, let $X_{n}$ be the cellulation of the 2-sphere with two
faces glued along an equatorial circle with $n$ vertices and $n$ edges. Let
$X=X_{1}$, and make it into a standard 2-complex by giving both faces area 1.
There is an $n$-sheeted map $\phi_{n}:X_{n}\to X$ that wraps the equator of
$X_{n}$ $n$ times around the equator of $X$. If we set the area of each face
of $X_{n}$ equal to $n$, then the map $\phi_{n}$ is a branched morphism.

Our goal is to understand a 2-complex $X$ by analysing a large set of maps
$Y\to X$, so it is of the utmost importance to constrain the kinds of maps we
allow.

######  Definition 2.4 (Branched coverings and immersions).

A branched map $\phi:Y\to X$ is a _branched immersion_ if it is locally
injective away from the midpoints of the 2-cells; equivalently, a branched
immersion is a map that is injective on the links of vertices. If $X$ and $Y$
are standard then $\phi$ is an _immersion_ , and is locally injective
everywhere.

If $\phi$ is a bijection on the links of vertices, then $\phi$ is a _branched
covering_. An immersion which is a branched covering is exactly the usual
topological notion of _covering map_.

Stallings famously explained how to study the subgroups of free groups by
folding maps of graphs [44, §5.4]. Stallings’ folding procedure provides an
algorithm to factor a map of finite graphs as a composition of a
$\pi_{1}$-surjection and an immersion. Our set-up makes it easy to extend
folding to 2-complexes: one just folds the 1-skeleta, and then identifies any
2-cells with the same image. The next lemma makes this precise.

######  Lemma 2.5 (Folding 2-complexes).

Every map of finite, connected, branched 2-complexes $\phi:Y\to X$ factors as

$Y\stackrel{{\scriptstyle\phi_{0}}}{{\to}}\bar{Y}\stackrel{{\scriptstyle\bar{\phi}}}{{\to}}X$

where $\phi_{0}$ is a surjection on fundamental groups and $\bar{\phi}$ is a
branched immersion.

Cousins of this lemma have already been used extensively [27, 28, 30, 29]. The
reader is referred to [49, Lemma 3.8] for a proof.

It would be convenient if we could always work with branched immersions.
However, sometimes it is necessary to _unfold_ in order to understand the
structure of a given complex, as the following simple example shows.

######  Example 2.6 (Folded surface).

Consider a compact surface $S$ equipped with a simplicial triangulation (i.e.
every edge and face has distinct vertices), and let $X$ be the result of
folding a single pair of adjacent edges of $S$ at a common vertex. Then there
are no branched immersions from closed surfaces to $X$.

In order to handle examples like this, we relax the definition of a branched
immersion slightly to an _essential map_. In related contexts, this definition
already played important roles in [48] and [29]. The idea is that folding an
essential map $Y\to X$ does not change any important information about $Y$.

######  Definition 2.7 (Essential map).

A map of branched 2-complexes $\phi:Y\to X$ is _essential_ if, after folding,
the resulting map $\phi_{0}:Y\to\bar{Y}$ satisfies the following conditions.

  1. (i)

The induced map on the 1-skeleton $\phi_{0}:Y_{(1)}\to\bar{Y}_{(1)}$ is a
homotopy equivalence.

  2. (ii)

When restricted to the interiors of faces, $\phi_{0}$ is a homeomorphism.
Equivalently, $\phi_{0}$ preserves area.

If $\phi$ is essential and, furthermore, $\bar{\phi}:\bar{Y}\to X$ is an
isomorphism, then $\phi$ is called an _essential equivalence_.

Note that the fold map $S\to X$ from Example 2.6 is essential, indeed an
essential equivalence. This illustrates an important principle: given a
2-complex $X$, there may be an essential equivalence $Y\to X$ such that the
structure of $Y$ is more apparent than the structure of $X$.

##  3 Group pairs

It is often convenient to think about 2-complexes in a more algebraic way,
provided by the following definition. Throughout this section, the 2-complex
$X$ is assumed to be connected.

######  Definition 3.1 (Group pair).

A _group pair_ $(F,\mathcal{A})$ consists of a finitely generated free group
$F$ together with a cofinite action of $F$ on a set $\mathcal{A}$ such that
every stabiliser is infinite cyclic.

There is a natural theory of group pairs, which it is not hard to develop on
its own terms. However, the point of this definition is that it captures some
of the underlying information of a 2-complex, and so we will develop the
theory in parallel with the corresponding properties of 2-complexes and the
maps between them.

######  Definition 3.2 (Group pair of a 2-complex).

Every finite, connected 2-complex $X$ has an _associated group pair_
$(F_{X},\mathcal{A}_{X})$, defined as follows. The free group $F_{X}$ is the
fundamental group of the 1-skeleton $X_{(1)}$. The boundaries of the faces
define a finite set $\\{\langle w_{1}\rangle,\ldots,\langle w_{n}\rangle\\}$
of infinite cyclic subgroups of $F_{X}$, each defined up to conjugacy. The set
$\mathcal{A}_{X}$ is then the disjoint union of the sets of left-cosets
$F_{X}/\langle w_{i}\rangle$, with the natural action by left multiplication.

######  Remark 3.3.

Usually $X$ is assumed to be concise. This means that the set $\\{\langle
w_{1}\rangle,\ldots,\langle w_{n}\rangle\\}$ is malnormal, and in this case
$\mathcal{A}_{X}$ can be identified with the disjoint union of the sets of
conjugates of the $\langle w_{i}\rangle$, with the natural action by
conjugation. We give this slightly more complicated definition in order to
handle 2-complexes that are not concise.

A _morphism_ of a group pairs $\phi:(F,\mathcal{A})\to(G,\mathcal{B})$
consists of a homomorphism $\phi:F\to G$ and a set map
$\phi:\mathcal{A}\to\mathcal{B}$ subject to the usual equivariance condition,
i.e. that $\phi(g.a)=\phi(g)\phi(a)$. Any map of 2-complexes $\phi:Y\to X$
gives rise to a natural map of the associated group pairs

$\phi_{*}:(F_{Y},\mathcal{A}_{Y})\to(F_{X},\mathcal{A}_{X})$

defined as follows. The group homomorphism $\phi_{*}:F_{Y}\to F_{X}$ is the
map induced on fundamental groups of 1-skeleta by $\phi$, but there is also an
equivariant map $\phi_{*}:\mathcal{A}_{Y}\to\mathcal{A}_{X}$: if $\phi_{*}$
sends the face corresponding to $\langle u_{i}\rangle$ to the face
corresponding to $\langle w_{j}\rangle$ then, after choosing conjugacy
representatives so that $\phi_{*}(u_{i})\in\langle w_{j}\rangle$, the map
$\mathcal{A}_{Y}\to\mathcal{A}_{X}$ is defined by

$\phi_{*}(h\langle w_{i}\rangle)=\phi_{*}(h)\langle u_{j}\rangle\,.$

A morphism of group pairs is said to be _injective_ if both the map of free
groups and the map of sets is injective. Perhaps surprisingly, this
corresponds exactly to the notion of essential map of 2-complexes defined in
the previous section. The following lemma [48, Lemma 1.12] characterises when
$\phi$ is essential in terms of $\phi_{*}$.

######  Lemma 3.4.

A map of finite, connected 2-complexes $\phi:Y\to X$ is essential if and only
if the induced map of associated group pairs

$\phi_{*}:(F_{Y},\mathcal{A}_{Y})\to(F_{X},\mathcal{A}_{X})\,,$

is injective.

The lemma shows that, up to essential equivalence, essential maps to $X$ from
finite, connected complexes $Y$ correspond naturally to subgroups $F_{Y}$ of
$F_{X}$ and $F_{Y}$-invariant cofinite subsets $\mathcal{A}_{Y}$ of
$\mathcal{A}_{X}$.

######  Remark 3.5.

It follows immediately from Lemma 3.4 that a composition of essential maps is
essential.

Two-complexes that are essentially equivalent in the sense of §2 have, in
particular, isomorphic group pairs. More generally, 2-complexes with
isomorphic group pairs are said to be _Nielsen equivalent_.

######  Definition 3.6.

An isomorphism of group pairs $(F,\mathcal{A})$ and $(H,\mathcal{B})$ consists
of a group isomorphism $\phi:F\to H$ and a $\phi$-equivariant bijection
$\mathcal{A}\to\mathcal{B}$. If the associated group pairs of 2-complexes $X$
and $Y$ are isomorphic, then $X$ and $Y$ are said to be _Nielsen equivalent_.

In particular, an essential equivalence of connected 2-complexes $f:Y\to X$ is
a morphism of 2-complexes that induces an isomorphism between the associated
group pairs.

For presentation complexes, Definition 3.6 coincides with the usual definition
of Nielsen equivalence of group presentations: $\langle a_{1},\ldots,a_{m}\mid
r_{1},\ldots,r_{n}\rangle$ and $\langle
a^{\prime}_{1},\ldots,a^{\prime}_{m}\mid
r^{\prime}_{1},\ldots,r^{\prime}_{n}\rangle$ are Nielsen equivalent if and
only if there is an isomorphism of free groups

$\alpha:\langle a_{1},\ldots a_{m}\rangle\to\langle a^{\prime}_{1},\ldots
a^{\prime}_{m}\rangle$

such that, up to permuting indices, $\alpha(r_{i})$ is conjugate to
$r^{\prime}_{i}$.

For branched 2-complexes, the area of the faces needs to be added to the data
of the pair.

######  Definition 3.7 (Branched group pair).

A _branched group pair_ consists of a group pair $(F,\mathcal{A})$ together
with an $F$-invariant map $\mathrm{Area}:\mathcal{A}\to\mathbb{R}_{\geq 0}$.
Two branched group pairs are said to be isomorphic if the underlying group
pairs $(F,\mathcal{A})$ and $(H,\mathcal{B})$ are isomorphic and the
associated bijection $\mathcal{A}\to\mathcal{B}$ preserves area.

It is easy to see that a branched 2-complex determines a branched group pair,
and so it makes sense to say that branched 2-complexes are Nielsen equivalent
if their associated branched group pairs are isomorphic.

##  4 Irreducible complexes

Two-dimensional complexes are too complicated to classify up to arbitrary
homotopy equivalence. However, sometimes they can be simplified, and the
following definition, which also played central roles in [48] and [29],
encapsulates various cases in which this is possible. The material in this
section is very similar to [29, §3.1–2]. Note that we do not assume that our
2-complexes are connected.

######  Definition 4.1 (Visibly reducible and unfoldable).

If a 2-complex $X$ satisfies any of the following three conditions then it is
said to be _visibly reducible_ :

  1. (i)

the one-skeleton of $X$ has a vertex of valence 0 or 1;

  2. (ii)

a vertex of $X$ is locally separating;

  3. (iii)

$X$ has a _free face_ , i.e. an edge that is hit exactly once by one face.

There is one further simplification move that plays an important role.

  1. (iv)

There is an essential equivalence $X^{\prime}\to X$ that is not an
isomorphism.

If $X$ satisfies condition (iv) but none of conditions (i)-(iii) then it is
said to be _unfoldable_. If $X$ satisfies none of conditions (i)-(iv) then it
is said to be _visibly irreducible_.

######  Remark 4.2.

Each of conditions (i)-(iv) in Definition 4.1 can be recognised from the
structure of the links of the vertices of $X$:

  1. (i)

the link of some vertex has at most one vertex;

  2. (ii)

the link of some vertex is disconnected;

  3. (iii)

the link of some vertex contains a vertex of valence 1;

  4. (iv)

the link of some vertex has a cut vertex.

In each case, there is a corresponding move that simplifies $X$.

In summary, every 2-complex is visibly reducible, or unfoldable, or visibly
irreducible. Unfolding increases the number of vertices and edges while
keeping the number of faces the same, and so it is not hard to see that a
finite complex can only be unfolded finitely many times, at which point it
will be either visibly reducible or visibly irreducible. Thus, the following
definition applies to all 2-complexes.

######  Definition 4.3 (Irreducible 2-complex).

A finite 2-complex $X$ is _irreducible_ if there is an essential equivalence
$X^{\prime}\to X$ such that $X^{\prime}$ is visibly irreducible. Otherwise,
$X$ is _reducible_.

Since it may be possible to unfold a 2-complex $X$ in several different ways,
it is _a priori_ possible that $X$ could, on the one hand, be essentially
equivalent to a visibly irreducible 2-complex $X^{\prime}$ and, on the other
hand, also be essentially equivalent to a visibly reducible 2-complex
$X^{\prime\prime}$. Using a famous lemma of Whitehead [46], it transpires that
irreducibility of a 2-complex can be characterised via algebraic properties of
the associated group pair. We will use the following convenient formulation
(cf. [29, Proposition 3.8]).

######  Lemma 4.4 (Whitehead’s lemma).

A finite 2-complex $X$ is reducible if and only if the associated group pair
$(F,\mathcal{A})$ satisfies one of the following conditions:

  1. (i)

$F$ is trivial;

  2. (ii)

$F$ is cyclic and $\\#\mathcal{A}=1$;

  3. (iii)

there is a non-trivial free splitting $F=A*B$ and each stabiliser of an
element of $\mathcal{A}$ is conjugate into either $A$ or $B$.

In particular, if $X$ is irreducible, then any unfolding sequence eventually
terminates in a visibly irreducible 2-complex.

Irreducible 2-complexes are very common. Indeed, even if $X$ is reducible, as
long as $\pi_{1}(X)$ is sufficiently complicated, a procedure of repeatedly
unfolding and simplifying will eventually produce a canonical irreducible
2-complex that maps to $X$. The following lemma is proved in [29, Lemma 3.10
and 3.14].

######  Lemma 4.5 (Irreducible cores).

Let $X$ be a connected, finite, standard 2-complex. If $\pi_{1}X$ is not free
and doesn’t split non-trivially as a free product, then there is an immersion
$\phi:\widehat{X}\to X$ such that $\widehat{X}$ is irreducible and $\phi$ is
an isomorphism of fundamental groups. Furthermore, any branched immersion from
an irreducible branched complex $Y\to X$ factors canonically through
$\widehat{X}$.

The complex $\widehat{X}$ is called the _irreducible core_ of $X$. In fact,
any finite complex $X$ can be factored into irreducible 2-complexes and simple
discs, where a _simple disc_ is a 2-complex $D$ with a single face and
homeomorphic to a disc.

######  Lemma 4.6 (Unfolding a 2-complex).

Let $X$ be any finite, connected, standard 2-complex. There is a finite,
connected graph $G$, a finite set of connected, irreducible 2-complexes
$X_{1},\ldots,X_{m}$, and a finite set of discs $D_{1},\ldots,D_{n}$ such that
there is an essential equivalence

$X^{\prime}=G\vee X_{1}\vee\ldots\vee X_{m}\vee D_{1}\vee\ldots\vee D_{n}\to
X$

for some choice of iterated wedge $X^{\prime}$ of the factors.

The proof is similar to the proofs of Lemmas 4.4 and 4.5, and left as an
exercise to the reader. The 2-complex $X^{\prime}$ is called an _unfolding_ of
$X$. We record one important consequence here.

######  Lemma 4.7 (Irreducible immersion).

Let $X$ be a connected, finite, standard 2-complex. Unless $X$ is homotopy
equivalent to a graph, there is an immersion $Y\to X$ with $Y$ irreducible.

###### Proof.

Consider an unfolding $X^{\prime}$ coming from Lemma 4.6. If $m\geq 1$ then we
may take $Y$ to be the result of folding the map $X_{1}\to X$. Otherwise,
$X^{\prime}$ is a obtained by wedging discs onto the graph $G$, and so is
homotopy equivalent to a graph, which implies that $X$ is too. ∎

Thus, as proposed in the introduction, we can study very general classes of
2-complexes $X$ via maps from irreducible 2-complexes. The reader should note,
however, that even the very weak hypothesis of Lemma 4.7 is not sharp, since
there are irreducible 2-complexes that are homotopic to a point.

######  Example 4.8.

Let $X$ be the presentation complex associated to the presentation $\langle
a,b\mid bab^{-1}a^{-2},b\rangle$. Since this is a balanced presentation of the
trivial group, it follows from the Hurewicz and Whitehead theorems that $X$ is
contractible. But the link of the unique vertex of $X$ is the complete graph
on 4 generators, so $X$ is irreducible by Remark 4.2.

Example 4.8 is the first of an infinite family of examples of contractible,
irreducible 2-complexes constructed by Miller and Schupp [35].

##  5 Surfaces

A finite 2-complex $X$ is said to be a _surface_ if its realisation is a
closed topological surface. Equivalently, $X$ is a surface if and only if the
link of every vertex is a circle. Since circles do not satisfy any of the
conditions of Remark 4.2, it follows that any surface is irreducible. Surfaces
provide some of the most important examples of irreducible complexes, and they
will play a special role.

Just as Whitehead’s lemma characterises the group pairs associated to
irreducible complexes, we will also need a result that characterises the
2-complexes that are Nielsen equivalent to surfaces. This is provided by the
following lemma; the proof follows a theorem of Culler [16].

######  Lemma 5.1 (Culler’s lemma).

If $X$ is a finite 2-complex which is Nielsen equivalent to a surface $S$ then
there is a (homeomorphic, but possibly combinatorially distinct) surface
$S^{\prime}$ and an essential equivalence $S^{\prime}\to X$.

###### Proof.

Let $N$ be a closed tubular neighbourhood of the 1-skeleton of $S$ in its
realisation. Then $N$ is homotopy equivalent to $S_{(1)}$ and the boundary
components of $N$ correspond exactly to the faces of $S$. Because $X$ and $S$
are Nielsen equivalent, the fundamental groups of the 1-skeleta $S_{(1)}$ and
$X_{(1)}$ are isomorphic, and because graphs are Eilenberg–Mac Lane spaces,
this isomorphism is induced by a homotopy equivalence $\phi:N\to X_{(1)}$.

Since the group isomorphism comes from a Nielsen equivalence it also respects
the faces, and $\phi_{*}$ sends the conjugacy classes of the cyclic subgroups
of $\pi_{1}(N)$ corresponding to the boundary components of $N$ bijectively to
the conjugacy classes of the cyclic subgroups of $\pi_{1}(X_{(1)})$
corresponding to the faces of $X$.

We now follow the proof of [16, Theorem 1.4] to endow the realisation of $S$
with a new 2-complex structure, so that $\phi$ becomes a morphism of
2-complexes. After a homotopy, we may take $\phi$ to be transverse to the
midpoints of edges of $X_{(1)}$. The preimages of these midpoints are then
disjoint unions of properly embedded circles and arcs in $N$. Furthermore, the
complementary regions map to neighbourhoods of vertices and so, since $\phi$
is a homotopy equivalence, every complementary region is a topological disc.

In particular, each embedded circle in the preimage bounds a disc, and so can
be removed by a homotopy of $\phi$ which reduces the number of components of
preimages. In the same way, we may also remove any arcs that are isotopic into
the boundary of $N$.

After these modifications, the union of the premages of the midpoints of edges
of $X_{(1)}$ is a collection of properly embedded arcs, which cut $N$ into
discs. The dual of this decomposition of $N$ is a graph $S^{\prime}_{(1)}$,
and $\phi$ naturally induces a morphism of graphs $\phi^{\prime}$ from
$S^{\prime}_{(1)}$ to $X_{(1)}$. Furthermore, $\phi^{\prime}$ is a homotopy
equivalence because $\phi$ was. Finally, we view the natural maps from the
boundary components of $N$ to $S^{\prime}_{(1)}$ as attaching maps to define a
2-complex $S^{\prime}$, and $\phi^{\prime}$ extends to a morphism of
2-complexes $S^{\prime}\to X$. We have already seen that $\phi^{\prime}$ is a
homotopy equivalence of 1-skeleta, and the construction also guarantees that
$\phi^{\prime}$ sends faces to faces bijectively, so $\phi^{\prime}$ is in
fact an essential equivalence. ∎

While Lemma 4.7 implies that there exist essential maps from irreducible
2-complexes to a 2-complex $X$ under very mild hypotheses on $X$, it is far
from obvious that there should be a similar result guaranteeing the existence
of essential maps from surfaces to general $X$. However, such a result exists,
and follows by combining Culler’s lemma with [48, Theorem F].

######  Theorem 5.2 (Essential maps from surfaces exist).

Every finite, irreducible complex $X$ admits an essential branched map $S\to
X$, where $S$ is a surface.

Surfaces are in some sense the simplest irreducible 2-complexes. The next
theorem offers one way of making this heuristic precise.

######  Theorem 5.3 (Surfaces are minimal among irreducible complexes).

A finite, irreducible 2-complex $X$ is a surface if and only if it satisfies
the following property: every essential map $\phi:Y\to X$, with $Y$ finite and
irreducible, is a finite-sheeted branched covering map.

###### Proof.

We start with the ‘only if’ direction. Suppose that $X$ is a surface, so every
link of a vertex in $X$ is a circle. Consider an essential map $\phi:Y\to X$,
where $Y$ is irreducible. If $\bar{\phi}:\bar{Y}\to X$ is the folded
representative then $\bar{Y}$ is also irreducible, by Whitehead’s lemma. Since
the folded representative is a branched immersion, it induces injective maps
on links; in particular, the link of each vertex of $\bar{Y}$ embeds into a
circle. But every proper, non-empty subgraph of a circle has a vertex of
valence either zero or one, which would imply that $\bar{Y}$ is visibly
reducible, a contradiction. Therefore, $\bar{\phi}$ is surjective on links and
hence a branched covering map. Finally, since circles do not have cut
vertices, $\bar{Y}$ is not unfoldable, and hence $Y=\bar{Y}$.

To prove the ‘if’ direction, suppose that $X$ satisfies the given property, so
that any essential map $Y\to X$ is a finite-sheeted branched covering map if
$Y$ is finite and irreducible. By Theorem 5.2, there is an essential map $S\to
X$ with $S$ of surface type. By the hypothesis the map $S\to X$ is a branched
covering map, so the induced maps on links are isomorphisms. Thus every link
of $X$ is also a circle, and so $X$ is also a surface. ∎

##  6 Curvature invariants

We are now ready to define our notions of curvature. The first definitions
adapt Euler characteristic to the context of branched complexes.

######  Definition 6.1 (Total and average curvature).

The _total curvature_ of a finite 2-complex $X$ is the quantity

$\tau(X):=\mathrm{Area}(X)+\chi(X_{(1)})$

where $\chi(X_{(1)})$ is the usual Euler characteristic of the 1-skeleton
$X_{(1)}$, namely $\\#V_{X}-\\#E_{X}$. The _average curvature_ of $X$ is

$\kappa(X):=\frac{\tau(X)}{\mathrm{Area}(X)}=1+\frac{\chi(X_{(1)})}{\mathrm{Area}(X)}\,.$

The total curvature is closely related to the Euler characteristic.

######  Remark 6.2.

If $X$ is standard then $\tau(X)$ is the Euler characteristic $\chi(X)$.
Furthermore, if $\phi:Y\to X$ is a branched map then each face $f$ has area
$m_{\phi}(f)\geq 1$, and so $\tau(Y)\geq\chi(Y)$.

On their own, the total and average curvatures cannot determine much about
$X$. The idea of our next invariants is to extremise these quantities over
complexes mapping to $X$.

######  Definition 6.3 (Irreducible curvature).

Let $X$ be a finite branched 2-complex, and let $\mathrm{Irred}(X)$ denote the
set of essential maps $Y\to X$, where $Y$ is any finite, irreducible, branched
2-complex. The quantity

$\rho_{+}(X):=\sup_{Y\in\mathrm{Irred}(X)}\kappa(Y)$

is called the _maximal irreducible curvature_ of $X$. Likewise, the _minimal
irreducible curvature_ $\rho_{-}(X)$ is defined to be the corresponding
infimum. By convention, $\rho_{\pm}(X)=\mp\infty$ if $\mathrm{Irred}(X)$ is
empty.

We saw in §4 that $\mathrm{Irred}(X)$ is usually non-empty; for instance,
Lemma 4.7 says that $\mathrm{Irred}(X)$ is non-empty unless $X$ is homotopy-
equivalent to a graph.

Definition 6.3 can be used to say that a 2-complex $X$ has _non-positive_
irreducible curvature (if $\rho_{+}(X)\leq 0$), _negative_ irreducible
curvature (if $\rho_{+}(X)<0$), _constant_ irreducible curvature
($\rho_{+}(X)=\rho_{-}(X)$), and so on. Since geometric group theorists are
traditionally most interested in upper bounds on curvature, we will be
especially concerned with $\rho_{+}(X)$.

Instead of extremising over essential maps from the relatively large class of
irreducible complexes, one may instead restrict attention to the smaller class
of (closed) surfaces, and this leads to another pair of invariants in the same
way.

######  Definition 6.4 (Surface curvature).

Let $X$ be a finite branched 2-complex, and let $\mathrm{Surf}(X)$ denote the
set of essential maps $Y\to X$, where $Y$ is a closed surface. Note that
$\mathrm{Surf}(X)\neq\varnothing$ by Theorem 5.2. The quantity

$\sigma_{+}(X):=\sup_{Y\in\mathrm{Surf}(X)}\kappa(Y)$

is called the _maximal surface curvature_ of $X$. Likewise, the _minimal
surface curvature_ $\sigma_{-}(X)$ is defined to be the corresponding infimum.
We again adopt the convention that $\sigma_{\pm}(X)=\mp\infty$ if
$\mathrm{Surf}(X)$ is empty.

Since a composition of essential maps is essential, Lemma 4.7 andTheorem 5.2
together guarantees that $\mathrm{Surf}(X)$ is non-empty unless $X$ is
homotopy equivalent to a graph.

As before, Definition 6.4 can be used to make sense of the ideas of a
2-complex $X$ having _non-positive_ surface curvature (if $\sigma_{+}(X)\leq
0$), _negative_ surface curvature (if $\rho_{+}(X)<0$) etc., and we will be
especially concerned with $\sigma_{+}(X)$, because of geometric group theory’s
traditional focus on upper curvature bounds.

##  7 Inequalities and first examples

We start with a few easy observations. Throughout this section we will assume
that $X$ is an irreducible branched complex. However, the irreducibility
hypothesis can usually be relaxed to the weaker assumption that
$\mathrm{Irred}(X)$ is non-empty.

The first result records some easy inequalities between the invariants.

######  Lemma 7.1 (Trivial inequalities).

If $X$ is a finite, irreducible, branched 2-complex then

$-\infty\leq\rho_{-}(X)\leq\sigma_{-}(X)\leq\sigma_{+}(X)\leq\rho_{+}(X)\leq
1\,.$

###### Proof.

The first inequality, that $-\infty\leq\rho_{-}(X)$, is vacuous. Since
surfaces are irreducible, there is a natural inclusion
$\mathrm{Surf}(X)\subseteq\mathrm{Irred}(X)$, whence
$\rho_{-}(X)\leq\sigma_{-}(X)$ and $\sigma_{+}(X)\leq\rho_{+}(X)$. By Theorem
5.2, $\mathrm{Surf}(X)\neq\varnothing$, so $\sigma_{-}(X)\leq\sigma_{+}(X)$.
Finally, for any essential $Y\to X$, we have

$\kappa(Y)=1+\frac{\chi(Y_{(1)})}{\mathrm{Area}(Y)}$

by definition. If $Y$ is irreducible then in particular the 1-skeleton
$Y_{(1)}$ is a core graph, so $\chi(Y_{(1)})\leq 0$ and $\kappa(Y)\leq 1$.
Taking the supremum over all $Y\in\mathrm{Irred}(X)$, the final inequality
follows, ∎

Eventually, we will sharpen these inequalities to prove Theorem A. The next
lemma records that the invariants are monotonic in the obvious way under
essential maps.

######  Lemma 7.2 (Monotonicity).

If $\psi:X^{\prime}\to X$ is an essential branched morphism of finite,
irreducible, branched 2-complexes then:

  1. (i)

$\rho_{+}(X^{\prime})\leq\rho_{+}(X)$ and
$\rho_{-}(X^{\prime})\geq\rho_{-}(X)$;

  2. (ii)

$\sigma_{+}(X^{\prime})\leq\sigma_{+}(X)$ and
$\sigma_{-}(X^{\prime})\geq\sigma_{-}(X)$.

###### Proof.

Since a composition of essential maps is essential, there is a natural map
$\psi_{*}:\mathrm{Irred}(X^{\prime})\to\mathrm{Irred}(X)$ obtained by
composing an essential map $Y\to X$ with $\psi$. Since
$\kappa(\psi_{*}(Y))=\kappa(Y)$, this proves item (i). Furthermore,
$\psi_{*}(\mathrm{Surf}(X^{\prime}))\subseteq\mathrm{Surf}(X)$, so item (ii)
also follows. ∎

When the essential map in question is a branched covering, we can say more.
The next lemma encodes the observation that average curvature is constant
under passing to branched coverings. Note that the restriction of any branched
covering map $\phi:X^{\prime}\to X$ to 1-skeleta is a covering map of graphs.
If $X$ is connected then this covering map has a degree, and we may define the
_degree_ of $\phi$ to be the degree of the covering map of 1-skeleta. In
particular, $\phi$ is said to be _finite sheeted_ if $\deg(\phi)$ is finite.

The next lemma shows that the quantities $\tau(X)$ and $\kappa(X)$ interact
well with finite-sheeted branched covering maps.

######  Lemma 7.3 (Finite-sheeted branched coverings).

If $\phi:X^{\prime}\to X$ is a finite-sheeted branched covering map of finite,
irreducible 2-complexes then $\tau(X^{\prime})=\deg(\phi)\tau(X)$ and
$\kappa(X^{\prime})=\kappa(X)$.

###### Proof.

The proof that

$\tau(X^{\prime})=\deg(\phi)\tau(X)$

is identical to the standard proof of the Riemann–Hurwitz theorem. Since
$\mathrm{Area}(X^{\prime})=\deg(\phi)\mathrm{Area}(X)$ trivially, it follows
that $\kappa(X^{\prime})=\kappa(X)$. ∎

It follows that our curvature bounds $\rho_{\pm}$ and $\sigma_{\pm}$ are also
invariant under passing to branched coverings.

######  Proposition 7.4 (Branched coverings and curvature bounds).

If $X^{\prime}\to X$ is a finite-sheeted branched covering map of finite,
irreducible 2-complexes then $\rho_{\pm}(X^{\prime})=\rho_{\pm}(X)$ and
$\sigma_{\pm}(X^{\prime})=\sigma_{\pm}(X)$.

###### Proof.

We prove the proposition for the irreducible curvatures; the proof for surface
curvatures is identical. By Lemma 7.2, $\rho_{-}(X)\leq\rho_{-}(X^{\prime})$
and $\rho_{+}(X)\geq\rho_{+}(X^{\prime})$, so it suffices to prove the reverse
inequalities.

Suppose now that $Y\in\mathrm{Irred}(X)$. Without loss of generality, we may
assume that $Y$ is visibly irreducible. Since branched immersions pull back
along essential maps [29, Lemma 3.13], there is a finite-sheeted branched
covering $Y^{\prime}\to Y$ and an essential map $Y^{\prime}\to X^{\prime}$.
Since $Y^{\prime}$ has the same links as the visibly irreducible complex $Y$,
it is itself visibly irreducible, so
$Y^{\prime}\in\mathrm{Irred}(X^{\prime})$. Now $\kappa(Y^{\prime})=\kappa(Y)$
by Lemma 7.3 and so, taking suprema or infima as appropriate, the result
follows. ∎

A little caution is needed when applying Proposition 7.4. If $X$ is standard
then $X^{\prime}$ is only standard if the map $X^{\prime}\to X$ is an
_unbranched_ covering.

The results discussed so far make it easy to compute the curvatures of our
first family of examples.

######  Example 7.5 (Surfaces).

Consider a closed surface $S$. By Theorem 5.3, $Y\in\mathrm{Irred}(S)$ if and
only if there is a finite-sheeted branched covering $Y\to S$, so
$\kappa(Y)=\kappa(S)$ by Lemma 7.3. Thus, by definition,

$\rho_{\pm}(S)=\kappa(S)$

and therefore $\sigma_{\pm}(S)=\kappa(S)$ by Lemma 7.1. In particular, if $S$
is standard and has one face, then all curvatures equal the Euler
characteristic $\chi(S)$.

In summary, surfaces have constant irreducible curvature. In fact, Theorem D
tells us that the equation $\rho_{+}(S)=\rho_{-}(S)$ is surprisingly close to
a characterisation of surfaces.

##  8 Nielsen equivalence

The curvature invariants defined here are too sensitive to be invariant under
homotopy equivalence. The irreducible, contractible 2-complex $X$ of Example
4.8 provides an absurd example, since

$\rho_{+}(X)\geq\kappa(X)=1/2$

but $X$ is homotopy equivalent to the 1-point complex $*$, which has
$\rho_{+}(X)=-\infty$.

Thus, $\rho_{+}$ is not invariant under homotopy equivalence, and indeed
similar reasoning tells us that neither are $\rho_{-}$ or $\sigma_{\pm}$.
However, they are invariant under a stronger equivalence relation. Recall from
§3 that two 2-complexes are _Nielsen equivalent_ if their associated group
pairs are isomorphic. The purpose of this section is to prove that the
quantities $\rho_{\pm}(X)$ and $\sigma_{\pm}(X)$ are invariant under Nielsen
equivalence. That is, we need to show that $\rho_{\pm}(X)$ and
$\sigma_{\pm}(X)$ only depend on the associated group pair
$(F_{X},\mathcal{A}_{X})$.

Since taking fundamental groups only makes good sense for connected complexes,
the first observation is that we may assume that every $Y\in\mathrm{Irred}(X)$
is connected.

######  Remark 8.1.

Let $Y$ be a finite branched 2-complex of positive area. If
$Y=Y_{1}\sqcup\ldots\sqcup Y_{n}$, then $\tau(Y)$ is the sum of the
$\tau(Y_{i})$, so

$\kappa(Y)=\sum_{i}t_{i}\kappa(Y_{i})$

where $0\leq t_{i}=\mathrm{Area}(Y_{i})/\mathrm{Area}(Y)\leq 1$. In
particular,

$\min_{i}\kappa(Y_{i})\leq\kappa(Y)\leq\max_{i}\kappa(Y_{i})$

and so, in the definitions of $\rho_{\pm}(X)$ and $\sigma_{\pm}(X)$, we may
restrict attention to connected $Y$ in $\mathrm{Irred}(X)$ or
$\mathrm{Surf}(X)$.

By Lemma 3.4, the branched group pairs of connected branched complexes in
$\mathrm{Irred}(X)$ correspond exactly to branched group pairs
$(F_{Y},\mathcal{A}_{Y})$ equipped with injective maps
$\phi_{*}:(F_{Y},\mathcal{A}_{Y})\to(F_{X},\mathcal{A}_{X})$. The next step is
therefore to compute $\kappa(Y)$ from the group pair
$(F_{Y},\mathcal{A}_{Y})$, which comes down to computing $\chi(Y_{(1)})$ and
$\mathrm{Area}(Y)$. The former is straightforward, since

$\chi(Y_{(1)})=1-\mathrm{rk}F_{Y}$

when $Y$ is connected.

For the latter, we need to associate an area to each $b\in\mathcal{A}_{Y}$,
which corresponds to some face $f$ of $Y$. The image
$a=\phi_{*}(b)\in\mathcal{A}_{X}$ has stabiliser conjugate to some $\langle
w_{j}\rangle$ which in turn corresponds to the face $\phi(f)$ of $X$. After
conjugating, we may assume that the stabiliser of $b$ is some $\langle
u_{i}\rangle$ and that $\phi_{*}(u_{i})\in\langle w_{j}\rangle$. The
multiplicity of $f$ can now be computed as the index

$m_{\phi}(f)=|\langle w_{i}\rangle:\langle\phi_{*}(u_{j})\rangle|$

so, by the compatibility between area and multiplicity, we may compute the
area of $f$ as

$\mathrm{Area}(f)=|\langle
w_{i}\rangle:\langle\phi_{*}(u_{j})\rangle|\mathrm{Area}(\phi(f))\,.$

Since $\mathrm{Area}(Y)$ is the sum of the areas of its faces, this shows that
$\mathrm{Area}(Y)$ can also be computed from the group pair.

Since Whitehead’s lemma (Lemma 4.4) characterises those group pairs
$(F_{Y},\mathcal{A}_{Y})$ that correspond to _irreducible_ complexes $Y$, it
follows that $\rho_{\pm}(X)$ are determined by the group pair
$(F,\mathcal{A})$ associated to $X$. Thus we have proved:

######  Proposition 8.2 (Nielsen invariance of irreducible curvatures).

If $X$ and $X^{\prime}$ are Nielsen equivalent 2-complexes, then
$\rho_{\pm}(X)=\rho_{\pm}(X^{\prime})$.

An identical argument substituting Culler’s lemma for Whitehead’s lemma shows
that $\sigma_{\pm}(X)$ also only depends on the group pair, yielding the
following:

######  Proposition 8.3 (Nielsen invariance of surface curvatures).

If two finite, irreducible 2-complexes $X$ and $X^{\prime}$ are Nielsen
equivalent, then $\sigma_{\pm}(X)=\sigma_{\pm}(X^{\prime})$.

##  9 Irrigid complexes

This section is devoted to studying a class of 2-complexes that slightly
generalise surfaces, and which played important roles in [47] and [48]. Here
we christen them _irrigid complexes_. Two important theorems are proved:
Theorem A, specifically the fact that $\rho_{-}=\sigma_{-}$, and Theorem D,
the classification of complexes of constant irreducible curvature.

######  Definition 9.1.

A _surface decomposition_ of a group pair $(F,\mathcal{A})$ is a graph of
groups $\mathcal{G}$ such that $\pi_{1}\mathcal{G}=F$ of the following form.
The edge groups are all infinite cyclic. The underlying graph is bipartite,
with vertices of the following two types.

  1. (i)

_Surface vertices_ $v$ have an associated compact surface with boundary
$\Sigma_{v}$, and the associated vertex group is identified with
$\pi_{1}(\Sigma_{v})$.

  2. (ii)

_Cyclic vertices_ have infinite-cyclic vertex groups.

For a surface vertex $v$, the incident edge groups are identified
isomorphically with the fundamental groups of the boundary components of
$\Sigma_{v}$. Some subset of the cyclic vertices, called the _peripheral_
vertices, is identified with the $F$-orbits of $\mathcal{A}$, and the vertex
group of a peripheral cyclic vertex is the corresponding stabiliser (which is
well-defined up to conjugacy).

A 2-complex $X$ is called _irrigid_ if the associated group pair of each
connected component admits a surface decomposition.

It is immediately clear that if $X$ is a surface then it is of surface type.
Baumslag–Solitar groups provide a more interesting class of examples.

######  Example 9.2.

Let $X$ be the standard presentation complex of the solvable Baumslag–Solitar
group

$BS(m,n)=\langle a,b\mid ba^{-n}b^{-1}a^{m}\rangle$

and let $r$ denote the relator $bab^{-1}a^{-n}$. The free group $a,b$ can be
viewed as the fundamental group of the following space. Let $\Sigma$ be a
3-holed sphere with boundary components $c_{0},c_{1},r$, and let $C$ be a
circle with fundamental group generated by a simple loop $a$. Then construct a
graph of spaces $\mathcal{X}$ by identifying $c_{0}$ with $a^{-m}$ and $c_{1}$
with $a^{n}$. This gluing introduces a stable letter $b$ in the fundamental
group of $\mathcal{X}$, and the fundamental group of $\mathcal{X}$ naturally
has presentation

$\langle a,b,c_{0},c_{1},r\mid
c_{0}c_{1}r\,,\,c_{0}a^{m}\,,c_{1}ba^{-n}b^{-1}\rangle\,.$

It follows that $\pi_{1}(\mathcal{X})$ is free on the generators $a,b$, while
$c_{0}=a^{-m}$, $c_{1}=ba^{n}b^{-1}$, and
$r=c_{1}^{-1}c_{0}^{-1}=ba^{-n}b^{-1}a^{m}=r$. Therefore, the graph-of-groups
decomposition for $F$ implicit in the construction of $\mathcal{X}$ is a
surface decomposition of $F$ relative to $r$. Since the only non-cyclic vertex
group is a surface, this shows that $BS(m,n)$ is irrigid.

In general, an algorithm of Cashen can be used to determine whether or not a
given pair is irrigid [12].

Although irrigid complexes are more general than surfaces, they can in
practice be replaced by branched maps from surfaces.

######  Lemma 9.3.

If $X$ is a finite, irreducible, irrigid complex, then there is a branched
surface $S$ with $\kappa(S)=\kappa(X)$ and an essential map $S\to X$.

###### Sketch proof.

Working componentwise, we may assume that $X$ is connected, so the associated
group pair $(F,\mathcal{A})$ has a surface decomposition $\mathcal{G}$ by
hypothesis. After replacing $X$ by a branched cover, we may assume that every
edge attaching map of $\mathcal{G}$ is an isomorphism by Marshall Hall’s
theorem [25].

For each cyclic vertex group of $\mathcal{G}$, choose a cyclic ordering on the
incident edges. Since $X$ is irreducible, each non-peripheral cyclic vertex
has at least two incident edges. Therefore, each edge is distinct from its
successor in the cyclic order. Consider the double cover
$X^{\prime}=X_{1}\sqcup X_{2}$ of $X$ that consists of the disjoint union of
two copies of $X$, corresponding to two copies $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ of $\mathcal{G}$. We now build a new graph of groups
$\mathcal{H}$ by deleting all of the cyclic vertices and, for each edge group
of $\mathcal{G}_{1}$, identifying it with the next edge group of
$\mathcal{G}_{2}$ in the relevant cyclic order.

The resulting graph of groups $\mathcal{H}$ consists of surfaces glued along
their boundary components and so, attaching 2-cells to each vertex group as
they were attached in the two copies of $\mathcal{G}$, it defines a group pair
$(H,\mathcal{B})$ of surface type. By construction, there is an essential
induced map of group pairs $(H,\mathcal{B})\to(F,\mathcal{A})$ so
$(H,\mathcal{B})$ corresponds to an essential map $Y\to X$ for some 2-complex
$Y$777This is where the fact that each edge group is distinct from its
successor in the cyclic order is used.. Since $(H,\mathcal{B})$ is of surface
type, there is a surface $S$ and elementary equivalence $S\to Y$ by Culler’s
lemma.

It remains to compute $\kappa(S)$. By Lemma 7.3, $\kappa(X)=\kappa(X_{1}\sqcup
X_{2})$. Since $\mathcal{H}$ is obtained from $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ by cutting and gluing infinite cyclic subgroups (which have
Euler characteristic zero), it follows that $\chi(Y_{(1)})=2\chi(X_{(1)})$.
Since each 2-cell of $X$ appears exactly twice in $X_{1}\sqcup X_{2}$ and
hence twice in $Y$, it follows that $\mathrm{Area}(Y)=2\mathrm{Area}(X)$.
Hence $\kappa(Y)=\kappa(X)$ and the result follows since $S\to Y$ is an
essential equivalence. ∎

The next lemma is a restatement of [48, Lemma 5.9], which in turn is based on
the main argument of [47].

######  Lemma 9.4.

Let $X$ be a finite, connected, irreducible, branched 2-complex. If $X$ is not
irrigid then there is $Y\in\mathrm{Irred}(X)$ such that $\kappa(Y)<\kappa(X)$.

With these results in hand, we can now easily prove two of the theorems from
the introduction.

###### Proof of Theorem A.

In the light of Lemma 7.1, it suffices to prove that

$-\infty<\rho_{-}(X)=\sigma_{-}(X)\,.$

The rationality theorem tells us that $\rho_{-}(X)$ is realised by some
$Y\in\mathrm{Irred}(X)$, so $\rho_{-}(X)=\kappa(Y)>-\infty$. Furthermore, $Y$
must be irrigid by Lemma 9.4, so Lemma 9.3 provides $S\in\mathrm{Surf}(Y)$
such that $\kappa(S)=\kappa(Y)=\rho_{-}(X)$. Now

$\sigma_{-}(X)\leq\sigma_{-}(Y)\leq\kappa(S)=\rho_{-}(X)$

by Lemma 7.2, and the result follows. ∎

Our second theorem is the classification of complexes with constant
irreducible curvature. Precisely, it asserts that every such 2-complex is
irrigid.888Strictly speaking, this is not quite a classification because the
necessary condition is not sufficient: there are irrigid 2-complexes with
$\rho_{+}(X)>\rho_{-}(X)$. But it is not difficult to determine which irrigid
2-complexes have constant irreducible curvature.

###### Proof of Theorem D.

If $X$ is not irrigid then Lemma 9.4 gives $Y\in\mathrm{Irred}(X)$ such that
$\kappa(Y)<\kappa(X)$. In particular,

$\rho_{-}(X)\leq\kappa(Y)<\kappa(X)\leq\rho_{+}(X)$

which contradicts the hypothesis that $\rho_{-}(X)=\rho_{+}(X)$. ∎

Since it plays an important role, it is also worth mentioning that the results
stated here also give a proof that $\mathrm{Surf}(X)$ is non-empty for
irreducible $X$ (i.e. Theorem 5.2), which is slightly different to the
original proof [48].

###### Proof of Theorem 5.2.

Let $X$ be finite and irreducible, so $\mathrm{Irred}(X)\neq\varnothing$ since
$X\in\mathrm{Irred}(X)$. By the rationality theorem, there is
$Y\in\mathrm{Irred}(X)$ such that $\kappa(Y)=\rho_{-}(X)$. By Lemma 9.4, $Y$
is irrigid, so there is $S\in\mathrm{Surf}(Y)$ by Lemma 9.3. The natural
embedding of $\mathrm{Surf}(Y)\to\mathrm{Surf}(X)$ now shows that
$\mathrm{Surf}(X)$ is also non-empty, as required. ∎

##  10 Relations to other invariants

Although $\sigma_{\pm}$ and $\rho_{\pm}$ appear to be new invariants, they are
related to various other invariants already in the literature. The purpose of
this section is to explain some of these relationships.

###  10.1 Deficiency

The deficiency of a presentation is an alternative normalisation of the Euler
characteristic, traditionally used in combinatorial group theory.

######  Definition 10.1 (Deficiency).

The _deficiency_ of a finite group presentation

$\langle a_{1},\ldots,a_{n}\mid r_{1},\ldots,r_{m}\rangle$

is the quantity $n-m$. Note that this is $\delta(X)=1-\chi(X)$, where $X$ is
the associated presentation complex. The _deficiency_ $\delta(G)$ of a group
$G$ is the maximum deficiency of a presentation for that group.

Presentations with deficiencies bounded below have been a popular topic
throughout the history of combinatorial group theory.999The Baumslag–Pride
theorem, that groups with $\delta(G)\geq 2$ are large, is perhaps the most
notable result in this direction [1]. One limitation is that no general
theorems can be proved about the subgroups of groups with deficiency bounded
below, since by taking free products any finitely presented group appears as a
free factor in a presentation with arbitrarily large deficiency. However,
upper bounds on irreducible curvature in turn impose a natural lower bound on
the deficiencies of the freely indecomposable finitely presented subgroups of
$G=\pi_{1}(X)$.

######  Remark 10.2.

Let $X$ be a finite, standard 2-complex. The inclusion of any non-free, freely
indecomposable, finitely presented subgroup $H$ of $G=\pi_{1}(X)$ is
represented by an immersion $Y\to X$, where $Y$ is irreducible [28, Lemma
4.2]. If $\rho_{+}(X)=0$ then $\tau(Y)\leq 0$ for every such $Y$, and so

$\delta(H)\geq\delta(Y)=1-\chi(Y)\geq 1-\tau(Y)\geq 1\,.$

Likewise, if $\rho_{+}(X)<0$ then $\tau(Y)<0$ for every such $Y$, and so
$\delta(H)\geq 2$.

This suggests that $\rho_{+}(X)$ should be thought of as a a ‘stabilised’
version of deficiency. The 2-complexes with non-positive irreducible curvature
provide good candidates for presentations of deficiency at least one about
which one can hope to prove useful theorems. Likewise, the 2-complexes with
negative irreducible curvature provide good candidates for presentations of
deficiency at least two to study. A few such theorems are collected in §12
below.

###  10.2 Primitivity rank

Primitivity rank is an invariant of words in free groups introduced by Puder
[40]. A word $w$ in a free group $F$ is _primitive_ if it is contained in a
minimal generating set, and otherwise is said to be _imprimitive_.

######  Definition 10.3 (Primitivity rank).

Let $F$ be a free group and $w\in F$ a non-trivial word. The quantity

$\pi(w):=\min\\{\mathrm{rk}(H)\mid w\mathrm{\,imprimitive\,in\,}H\\}$

is the _primitivity rank_ of $w$. By convention, if $w$ is primitive in $F$
(and hence in all its subgroups) then $\pi(w)=\infty$.

Puder and Parzanchevski related the primitivity rank to the behaviour of
random word maps on symmetric groups [41], while Louder and the author related
it to the subgroup structure of 1-relator groups [30]101010See §11.3 for
further discussion of one-relator groups..The connection with one-relator
groups stems from the observation that the primitivity rank is also related to
an extremal value of the total curvature, when restricted to immersed
2-complexes of area 1.

######  Lemma 10.4.

Let $F$ be a free group, let $w\in F$ be a non-trivial element, and let $X$ be
the natural presentation complex of the one-relator group $F/\llangle
w\rrangle$. Then

$2-\pi(w)=\max\\{\tau(Y)\mid Y\in\mathrm{Irred}(X)\,,\,\mathrm{Area}(Y)=1\\}$

###### Proof.

If $H\leq F$ realises the minimum in the definition of $\pi(w)$ then the
associated immersion of core graphs $Y_{(1)}\to X_{(1)}$ carries a lift
$\hat{w}$ of $w$. Since $H$ cannot split freely relative to $w$, the complex
$Y$ constructed by attaching a face along $\hat{w}$ to $Y_{(1)}$ is
irreducible, by Whitehead’s lemma. Since

$\tau(Y)=2-\mathrm{rk}H=2-\pi(w)$

this establishes that the left hand side of the desired equation is no greater
than the right hand side.

To prove the reverse inequality, it remains to show that any other
$Z\in\mathrm{Irred}(X)$ with $\mathrm{Area}(Z)=1$ has $\tau(Z)\leq\tau(Y)$.
Indeed, the 1-skeleton $Z_{(1)}$ defines a subgroup $K$ of $F$ containing $w$
imprimitively, again by Whitehead’s lemma. But $\tau(Z)>\tau(Y)$ implies that
$\mathrm{rk}(K)<\mathrm{rk}(H)$, because
$\mathrm{Area}(Z)=\mathrm{Area}(Y)=1$, contradicting the hypothesis that $H$
realises the minimum. ∎

The lemma quickly implies a relationship between the primitivity rank and the
maximal irreducible curvature.

######  Lemma 10.5 (Primitivity rank and $\rho_{+}(X)$).

If $X$ is the presentation complex of the one-relator group $F/\llangle
w\rrangle$ then

$2-\pi(w)\leq\rho_{+}(X)\,.$

###### Proof.

If $\mathrm{Area}(Y)=1$ then $\kappa(Y)=\tau(Y)$, so Lemma 10.4 shows that
$2-\pi(w)=\kappa(Y)$ for some $Y\in\mathrm{Irred}(X)$. The result is then
immediate. ∎

This result provides useful estimates of $\rho_{+}$ because the primitivity
rank can often be computed by hand for small examples; see, for instance, [30,
Example 1.4].111111More generally, algorithms to compute $\pi(w)$ are
described in [41, Appendix A] and [30, Lemma 6.4].

Lemma 10.4 suggests that the quantity $\rho_{+}(X)$ can be thought of as
stabilised version of primitivity rank, in analogy with stabilising commutator
length to give stable commutator length.121212See §10.3 for more on stable
commutator length. We therefore propose the following definition.

######  Definition 10.6.

Let $F$ be a free group and $w\in F$ be a non-trivial element. The _stable
primitivity rank_ of $w$ is defined to be

$s\pi(w):=1-\rho_{+}(X)$

where $X$ is the natural presentation complex associated to the one-relator
group $F/\llangle w\rrangle$.131313The reader may reasonably wonder why
$s\pi(w)$ is defined to be $1-\rho_{+}(X)$, rather than $2-\rho_{+}(X)$. This
is because $s\pi(w)$ should be thought of as a (suitably normalised) Euler
characteristic of a graph, not of a 2-complex, so we should subtract the term
coming from the face of $X$.

A much stronger improvement of Lemma 10.4 was proved by Louder and the author
in [30, Lemma 6.10].

######  Theorem 10.7.

If $X$ is the presentation complex of the one-relator group $F/\llangle
w\rrangle$ then

$2-\pi(w)=\max_{Y\in\mathrm{Irred}(X)}\tau(Y)\,.$

This theorem provides a useful method to prove that $\rho_{+}(X)$ is negative
– see, for instance, Examples 11.39 below.

###  10.3 Stable commutator length

Let $w$ be a non-trivial element of a free group $F$ as in the previous
section, but this time we assume that $w$ is in the commutator subgroup
$[F,F]$. Recall the standard definitions of commutator length and its stable
cousin.

######  Definition 10.8 (Commutator length and stable commutator length).

The _commutator length_ of $w$, $\mathrm{cl}(w)$, is the minimal positive
integer $n$ such that $w$ is a product of $n$ commutators. The quantity

$\mathrm{scl}(w):=\inf_{n\in\mathbb{N}}\frac{\mathrm{cl}(w)}{n}$

is the _stable commutator length_ of $w$. (In fact this makes sense for any
group.)

There are two important results about stable commutator length in a free group
$F$. The Duncan–Howie theorem asserts that $\mathrm{scl}(w)\geq 1/2$ unless
$w$ is a proper power [17]. Calegari’s rationality theorem asserts that
$\mathrm{scl}(w)\in\mathbb{Q}$, and furthermore there is an algorithm to
compute it [10]. In fact, this algorithm is effective, and has been
implemented [11].

As usual, we may consider a standard presentation complex $X$ for
$F/\llangle{w}\rrangle$, with a single face $f$. The next lemma documents a
relationship between the surface curvature of $X$ and the stable commutator
length.

######  Lemma 10.9 (Surface curvature and stable commutator length).

For $w\in[F,F]$ and $X$ the standard presentation complex of
$F/\llangle{w}\rrangle$,

$1-2\mathrm{scl}(w)\leq\sigma_{+}(X)\,.$

The lemma will follow from the stronger Proposition 10.12 below. The point of
Lemma 10.9 is that the quantity $1-2\mathrm{scl}(w)$ can be thought of as an
_oriented_ version of the surface curvature for the complex $X$. In this
oriented version the areas of the faces of surfaces mapping to $X$ are counted
with signs. Therefore, the oriented area is necessarily smaller than the
unoriented area, which leads to the stated inequality.

From this point of view, the Duncan–Howie theorem asserts that the oriented
surface curvature of a one-relator complex is always non-positive (cf. the
stronger Theorem 11.18 below), while Calegari’s theorem asserts that the
oriented surface curvature is rational and computable (cf. the rationality
theorem [49]).

In fact, the above discussion can be generalised beyond the one-relator
context, to an arbitrary oriented 2-complex.

######  Definition 10.10 (Orientation).

An _orientation_ on a branched 2-complex $X$ is a cellular 2-cycle

$c=\sum_{f\in F(X)}c_{f}.f$

such that $|c_{f}|=\mathrm{Area}(f)$ for each face $f$. An _oriented
2-complex_ is a 2-complex $X$ equipped with an orientation $c$.

Since $c$ is a 2-cycle it defines a non-trivial element of
$H_{2}(X,\mathbb{R})$, and also a non-trivial element of the second relative
homology of the 1-skeleton $X_{(1)}$ relative to the faces. In particular,
$\partial c$ is a boundary in the graph $X_{(1)}$, and so we can define the
_stable commutator length_ of $\partial c$. We recall the definition below,
and refer the reader to Calegari’s monograph [9] for an authoritative
discussion.

######  Definition 10.11 (Stable commutator length).

Let $\mathrm{Admiss}(c)$ be the set of continuous maps of compact, oriented
surfaces $S\to X_{(1)}$ (not necessarily connected, but without spherical
components) such that

$\partial S\mapsto\deg(S)\partial c\,,$

where $\deg(S)$ is a positive real number. Then

$\mathrm{scl}(c):=\inf_{S\in\mathrm{Admiss}(c)}-\frac{\chi(S)}{2\deg(S)}\,.$

In the case where $X$ is the presentation complex of a one-relator group
$F/\llangle w\rrangle$, this coincides with the usual definition of stable
commutator length given above [9, Proposition 2.10]. We can now phrase a
relationship between stable commutator length and surface curvatures for all
oriented 2-complexes.

######  Proposition 10.12.

Let $c$ be an orientation on a branched 2-complex $X$. Then

$\sigma_{-}(X)\leq
1-\frac{2\mathrm{scl}(c)}{\mathrm{Area}(X)}\leq\sigma_{+}(X)$

###### Proof.

By Calegari’s rationality theorem, there is $S\in\mathrm{Admiss}(c)$ such that

$\mathrm{scl}(c)=-\frac{\chi(S)}{2\deg(S)}\,.$

Calegari also shows that $S$ can be taken to be _monotone_ , meaning that the
orientation on $\partial S$ restricted from the orientation on $S$ coincides
with the orientation on $\partial S$ pulled back from $c$ [9, Proposition
2.13]. By capping off the boundary components, the surface $S$ defines a
closed surface $\widehat{S}\to X$ with
$\mathrm{Area}(\widehat{S})=\deg(S)\mathrm{Area}(X)$. Note also that

$\displaystyle\kappa(\widehat{S})$ $\displaystyle=$ $\displaystyle
1+\frac{\chi(S)}{\mathrm{Area}(\widehat{S})}$ $\displaystyle=$ $\displaystyle
1-\frac{2\deg(S)\mathrm{scl}(c)}{\deg(S)\mathrm{Area}(X)}$ $\displaystyle=$
$\displaystyle 1-\frac{2\mathrm{scl}(c)}{\mathrm{Area}(X)}\,.$

Such a surface $S$ is called _extremal_ and, by a lemma of Calegari [8, Lemma
2.7], $\widehat{S}$ is essential. In particular,
$\widehat{S}\in\mathrm{Surf}(X)$, and so

$\sigma_{-}(X)\leq\kappa(\widehat{S})\leq\sigma_{+}(X)$

which gives the result. ∎

Lemma 10.9 follows immediately by specialising to the case of a single face of
area 1. Again, the moral is that a suitable normalisation of stable commutator
length can be thought of as an oriented version of the surface curvature.

Recall that the Duncan–Howie theorem tells us that every one-relator complex
has non-positive oriented surface curvature. It remains an open question
whether or not every 2-boundary $c$ in a free group satisfies
$\mathrm{scl}(c)\geq 1/2$. However, from our point of view, the natural
threshold corresponding to non-positive oriented surface curvature for an
oriented complex is

$\mathrm{scl}(c)\geq\frac{\mathrm{Area}(X)}{2}\,,$

and some oriented 2-complexes do indeed have positive oriented surface
curvature: for instance, any cellulation of the 2-sphere provides an example.

Since stable commutator length can be computed in practice, Proposition 10.12
offers a practical way to obtain lower bounds on $\sigma_{+}$ and upper bounds
on $\sigma_{-}$.

##  11 Examples

So far, surfaces have provided the only classes of examples that we have
considered. The goal of this section is to collect a wider variety of
examples, which illustrate the various possibilities that can occur. In the
first few sections, however, we collect techniques for computing the
invariants.

###  11.1 Angle structures and $CAT(\kappa)$ spaces

A more traditional way to impose curvature restrictions on a 2-complex $X$ is
to use an _angle structure_ , namely a path metric on the link of each vertex.
Often, this angle structure is induced from a path metric on the whole of $X$,
where each edge of each link is assigned the angle of the corresponding
corner. Gromov’s link condition implies that a Euclidean (respectively,
hyperbolic) polygonal metric is locally $CAT(0)$ (respectively, locally
$CAT(-1)$) if and only if, in the induced angle structure, the link of every
vertex has systole at least $2\pi$ [6, Lemma II.5.6].

An angle structure also makes it possible to impose bounds on the average
curvature, following ideas of Wise [51].

######  Definition 11.1.

Consider a finite, irreducible 2-complex $X$, and suppose that $X$ is equipped
with an angle structure. Let $f$ be a face of $X$ which is an $n(f)$-gon,
meaning that its boundary is a circle with $n(f)$ edges, and with total
interior angle $\alpha(f)$. The _curvature_ of $f$ is the quantity

$c(f):=2\pi\mathrm{Area}(f)-\pi n(f)+\alpha(f)\,.$

We say that the angle structure is _non-positively curved_ (respectively,
_negatively curved_) if:

  1. (i)

the link of every vertex has systole at least $2\pi$; and

  2. (ii)

$c(f)\leq 0$ for each face $f$ (respectively, $c(f)<0$ for each face $f$).

This definition of the curvature of a face becomes easier to interpret with
some input from the Gauss–Bonnet theorem for geodesic polygons.

######  Lemma 11.2 (Riemannian Gauss–Bonnet).

Consider a face $f$ with $\mathrm{Area}(f)=1$. If $f$ is endowed with the
angle structure of a geodesic polygon $P$ in some Riemannian surface, then

$c(f)=\int_{P}\kappa_{\mathrm{Gauss}}.dA$

where $\kappa_{\mathrm{Gauss}}$ is the Gauss curvature.

###### Proof.

The Gauss–Bonnet theorem for geodesic polygons asserts that

$\int_{P}\kappa_{\mathrm{Gauss}}.dA=\alpha(f)-(n(f)-2)\pi$

and the result follows immediately by the definition of $c(f)$. ∎

In particular, the angle structures coming from locally $CAT(0)$ Euclidean
structures are non-positively curved, and the angle structures coming from
locally $CAT(-1)$ hyperbolic structures are negatively curved. The use of this
definition is in the following estimate, a combinatorial version of the
Gauss–Bonnet theorem.

######  Lemma 11.3 (Combinatorial Gauss–Bonnet).

Let $S$ be a finite branched surface equipped with an angle structure. If the
total angle around every vertex is at least $2\pi$ then

$2\pi\tau(S)\leq\sum_{f\in F_{S}}c(f)\,.$

In particular, if the angle structure is non-positively curved then
$\tau(S)\leq 0$. If the angle structure is negatively curved then $\tau(S)<0$.

###### Proof.

Since the link of each vertex has total angle at least $2\pi$, the number of
vertices of $X$ is at most the sum of the interior angles across $X$ divided
by $2\pi$. Hence

$\displaystyle\sum_{f\in F_{X}}c(f)$ $\displaystyle=$ $\displaystyle\sum_{f\in
F_{X}}2\pi\mathrm{Area}(f)-\pi n(f)+\alpha(f)$ $\displaystyle=$ $\displaystyle
2\pi\mathrm{Area}(X)-2\pi\\#E_{X}+\sum_{f\in F_{X}}\alpha(f)$
$\displaystyle\geq$ $\displaystyle
2\pi\mathrm{Area}(X)-2\pi\\#E_{X}+2\pi\\#V_{X}$ $\displaystyle=$
$\displaystyle 2\pi\tau(X)$

as required. ∎

Furthermore, the curvature of a face interacts well with branched maps.

######  Remark 11.4.

Suppose that $\phi:Y\to X$ is a branched map, and that $f$ is a face of $Y$.
If the image $\phi(f)$ has $n$ sides then $f$ has $m_{\phi}(f)n$ sides;
likewise, if $\alpha$ is the total interior angle of $\phi(f)$ then
$m_{\phi}(f)\alpha$ is the total interior angle of $f$. Since area is also
multiplicative, it follows that

$c(f)=m_{\phi}(f)c(\phi(f))\\\ $

by definition.

As a result, the curvature of a face provides an upper bound on the surface
curvature of a standard 2-complex.

######  Proposition 11.5 (Face curvature bounds surface curvature).

Let $X$ be a standard 2-complex equipped with an angle structure. If the link
of every vertex has systole at least $2\pi$ then

$2\pi\sigma_{+}(X)\leq\max_{f\in F_{X}}c(f)\,.$

###### Proof.

Consider an essential map from a surface $S\to X$. Since $S\to X$ is
essential, links of $S$ map to immersed loops in links of $X$, and in
particular are of length at least $2\pi$. Applying Lemma 11.3 and Remark 11.4,
we get

$\displaystyle 2\pi\tau(S)$ $\displaystyle\leq$ $\displaystyle\sum_{f\in
F_{S}}c(f)$ $\displaystyle=$ $\displaystyle\sum_{f\in
F_{S}}m_{\phi}(f)c(\phi(f))$ $\displaystyle\leq$
$\displaystyle\mathrm{Area}(S)\max_{f\in F_{X}}c(f)$

and the result follows. ∎

In particular, a locally $CAT(0)$ Euclidean 2-complex has non-positive surface
curvature.

######  Corollary 11.6 (Surface curvature of locally $CAT(0)$ 2-complexes).

If $X$ is a finite 2-complex equipped with a non-positively curved angle
structure (for instance, if $X$ is a locally CAT(0) Euclidean complex), then
$\sigma_{+}(X)\leq 0$.

Likewise, locally $CAT(-1)$ hyperbolic 2-complexes have negative surface
curvature.

######  Corollary 11.7 (Surface curvature of locally $CAT(-1)$ 2-complexes).

If $X$ is a finite 2-complex equipped with a negatively curved angle structure
(for instance, if $X$ is a locally CAT(-1) hyperbolic complex), then
$\sigma_{+}(X)<0$.

The proposition has one more consequence which is very useful in practice,
because it depends only on combinatorial information. By the _girth_ of a
graph we mean the combinatorial length of the shortest non-trivial loop.

######  Corollary 11.8 (Girths of links).

Let $X$ be a standard 2-complex in which every face has $n$ sides. Then

$\sigma_{+}(X)\leq 1-\frac{n}{2}\left(1-\frac{2}{\lambda}\right)$

where $\lambda$ is the smallest girth of the link of a vertex in $X$.

###### Proof.

Equip $X$ with an angle structure in which every edge of every link has length
$2\pi/\lambda$. Then the systole of every link is at least $2\pi$, and every
face has curvature

$c(f)=2\pi-\pi
n+2\pi\frac{n}{\lambda}=2\pi\left(1-\frac{n}{2}\left(1-\frac{2}{\lambda}\right)\right)$

and the result follows from Proposition 11.5. ∎

######  Remark 11.9 (Wise’s sectional curvature).

In Wise’s terminology, the 2-complexes considered in this section are of non-
positive or negative _planar sectional curvature_. In [51, Definition 2.4],
Wise defines a stronger notion of (non-planar) _sectional curvature_. In our
terminology, his arguments show that a 2-complex $X$ with an angle structure
of non-positive sectional curvature has $\rho_{+}(X)\leq 0$, while an angle
structure of negative sectional curvature would guarantee that
$\rho_{+}(X)<0$.

###  11.2 Small-cancellation and random complexes

Small-cancellation conditions are a useful combinatorial method of imposing
curvature-like constraints on group presentations; see the book of Lyndon and
Schupp for an authoritative treatment [32, Chapter V].

As in the proof of Lemma 2.5, a 2-complex $X$ is characterised by a map
$S_{X}\to X_{(1)}$, the attaching map of the boundaries of the 2-cells. A
_piece_ of $X$ is an off-diagonal component of the fibre product
$S_{X}\times_{X_{(1)}}S_{X}$. A 2-complex satisfies the $C(p)$ _small-
cancellation condition_ if no face of $X$ is a union of fewer than $p$
pieces.141414See the book of Lyndon and Schupp for a thorough treatment of
small-cancellation theory [32]. It is well known that the $C(6)$ condition can
be thought of as a form of non-positive curvature, while the $C(7)$ condition
can be thought of as a form of negative curvature. These heuristics fit
precisely into the framework of surface curvature.

######  Proposition 11.10 ($C(6)$ complexes have non-positive surface
curvature).

If $X$ is a finite, irreducible, standard, $C(6)$ complex then
$\sigma_{+}(X)\leq 0$.

###### Proof.

It suffices to prove that

$\tau(Y)\leq 0$

whenever $\phi:Y\to X$ is an essential map from a surface.

For a face $f$ of $Y$, let $n(f)$ be the number of vertices of $Y_{(1)}$
adjacent to $f$ of valence at least 3. Consider the natural graph structure on
$Y_{(1)}$ obtained by deleting the vertices of valence 2. The number of edges
is then exactly

$\sum_{f\in F_{Y}}n(f)/2$

while the number of vertices is at most

$\sum_{f\in F_{Y}}n(f)/3\,,$

so

$\tau(Y)\leq\sum_{f\in F_{Y}}\mathrm{Area}(f)-n(f)/2+n(f)/3=\sum_{f\in
F_{Y}}\mathrm{Area}(f)-n(f)/6\,.$

But the area of and $f\in F_{Y}$ is $m_{\phi}(f)$ because $X$ is standard, and
$n(f)\geq 6m_{\phi}(f)$ because $X$ is $C(6)$, so

$\tau(Y)\leq\sum_{f\in F_{Y}}m_{\phi}(1-6/6)=0$

as required. ∎

In exactly the same way, we obtain a similar result for $C(7)$ complexes.

######  Proposition 11.11 ($C(7)$ complexes have negative surface curvature).

If $X$ is a finite, irreducible, standard $C(7)$ complex then
$\sigma_{+}(X)<0$.

Any $C(7)$ presentation complex $X$ with at least as many relators as
generators now provides an example with $\sigma_{+}(X)<0$ but $\rho_{+}(X)>0$.
For an especially large class of examples, one can use random presentations in
the few-relator model.

######  Example 11.12 (Random few-relator complexes).

Consider a randomly chosen presentation in the few-relator model with $n$
generators and $m$ relators of length $l$. The probability that this
presentation is $C(7)$ tends to 1 as $l\to\infty$, and so the standard
presentation complex also satisfies $\sigma_{+}(X)<0$ with high probability.
On the other hand, $\chi(X)=1-n+m$ and the probability that $X$ is not visibly
irreducible tends to 0 as $l\to\infty$. Therefore, with probability tending to
1,

$\rho_{+}(X)\geq\frac{\tau(X)}{\mathrm{Area}(X)}=\frac{1-n+m}{m}=1+(1-n)/m>0$

unless $m<n$.

This provides a rich class of examples of complexes with negative surface
curvature but positive irreducible curvature.

Since the inequality used in Example 11.12 only provides a lower bound, it
remains an interesting problem to estimate the irreducible curvatures of few-
relator complexes with $m<n$. The following conjecture represents the most
optimistic possibility.

######  Conjecture 11.13 (Irreducible curvature bounds of few-relator
complexes).

If $X$ is a random few-relator complex as in Example 11.12, then the
probability that $\rho_{+}(X)=\kappa(X)$ tends to $1$ as $l\to\infty$. In
particular, with high probability, $\rho_{+}(X)<0$ if $m<n-1$ and
$\rho_{+}(X)=0$ if $m=n-1$.

The _density model_ of random groups, in which $m$ is proportional to
$(2n-1)^{dl}$ for some density $0<d<1$, is also much-studied, and the
resulting group is infinite and hyperbolic with high probability if $d<1/2$
[20]. These presentation complexes are irreducible and have positive Euler
characteristic by definition, so certainly $\rho_{+}(X)>0$. However, a random
complex satisfies $C(7)$ with high probability at density $d<1/12$ [37], so
such a complex $X$ does have negative surface curvature, and it is likely that
this estimate extends to densities less than $1/2$.

######  Conjecture 11.14 (Surface curvature of random complexes in the density
model).

For a random presentation complex $X$ in the density model at density $d<1/2$,
the probability that $\sigma_{+}(X)<0$ tends to $1$ as $l\to\infty$.

###  11.3 One-relator groups

Groups with a single defining relation were much studied in the early years of
combinatorial group theory. According to Chandler and Magnus, Dehn
hypothesised a geometric solution to their word problem [13, p. 114]. However,
Magnus’ eventual solution was combinatorial [33], and indeed none of the
curvature conditions most commonly used in geometric group theory – Gromov-
hyperbolicity, automaticity, $CAT(0)$ metrics etc – can apply to all one-
relator groups, since they all imply a quadratic isoperimetric inequality.

In 2003, Wise introduced notions of _non-positive_ and _negative immersions_
[50, 51] as tools to attack certain open problems about one-relator groups,
especially a famous question, due to Baumslag, asking whether or not one-
relator groups are coherent [3].151515See [52] for a survey of the current
state of knowledge about coherence. These ideas have been developed over the
last decade in papers by Helfer–Wise [21] and Louder and the author [27, 30],
culminating in resolutions of Baumslag’s question for one-relator groups with
torsion [28, 53] and 2-free one-relator groups [29]. These developments can be
interpreted as the beginning of a geometric theory of one-relator groups. In
this section, we will see that this new geometric theory of one-relator groups
can also be interpreted using maximal irreducible curvature.

Consider the standard presentation complex $X$ for the presentation

$\langle a_{1},\ldots,a_{n}\mid w\rangle$

of a one-relator group $G$. Recall that $G$ is torsion-free if and only if the
relator $w$ is not a proper power [32, Proposition 5.17]. Note also that, as
long as $n>1$, the standard presentation complex $X$ is irreducible if and
only if $w$ is not conjugate into a proper free factor.

The following theorem, proved independently by Helfer–Wise and by Louder and
the author, can be thought of as the first proof of an upper bound on
irreducible curvature [21, 27].

######  Theorem 11.15 (Non-positive irreducible curvature for one-relator
groups).

Let $X$ be the standard presentation complex of a one-relator group $G$, and
suppose that $X$ is irreducible. If $G$ is torsion-free then $\rho_{+}(X)\leq
0$.

The theorem gives an exact computation of $\rho_{+}$ for two-generator, one-
relator groups.

######  Corollary 11.16.

Let $X$ be the standard presentation complex of a two-generator, one-relator
group. If $X$ is irreducible and $G$ is torsion-free then $\rho_{+}(X)=0$.

###### Proof.

Since the presentation has two generators,

$\tau(X)=\chi(X)=2-2=0$

so the identity map realises the upper bound from the theorem. ∎

Theorem 11.15 also implies that $\sigma_{+}(X)\leq 0$ by Lemma 7.1. This
result contravenes the standard heuristics of geometric group theory. For
instance, it is usually posited that any reasonable notion of non-positive
curvature for a group $G$ should imply that $G$ satisfies a quadratic
isoperimetric inequality. However, there are famously Baumslag–Solitar groups
with exponential Dehn function [5, Exercise 7.2.11], and yet their
presentation complexes are non-positively curved in our sense.

######  Example 11.17 (Baumslag–Solitar groups).

The standard presentation complex $X$ of the $(m,n)$-Baumslag–Solitar group

$BS(m,n)\cong\langle a,b\mid ba^{m}b^{-1}a^{-n}\rangle$

has $\rho_{+}(X)=0$. Indeed, $\rho_{+}(X)\leq 0$ by Theorem 11.15, but
$\chi(X)=0$ so the upper bound is realised.

One-relator groups include many examples that are even more pathological, such
as the Baumslag–Gersten group $BG=\langle a,b\mid
b^{a}a(b^{a})^{-1}a^{-2}\rangle$ [2, 18]. From our perspective, all of these
are non-positively curved as long as they are torsion-free. (The case with
torsion is addressed in Theorem 11.19 below.)

The question of negative curvature for one-relator groups is considerably more
delicate. In Theorem 12.8 below, we shall see that if $\rho_{+}(X)<0$ and
$G=\pi_{1}(X)$ then every 2-generator subgroup of $G$ is free. The results of
Louder and the author [30, 29] show that this is the only obstruction to
negative irreducible curvature for one-relator groups.

######  Theorem 11.18 (Torsion-free one-relator groups with negative
irreducible curvature).

If $X$ is the standard presentation complex of a torsion-free one-relator
group $G$ and every two-generator, one-relator subgroup of $G$ is free then
$\rho_{+}(X)<0$.

In [30], these one-relator groups are said to have _negative immersions_.

Note that the standard presentation complexes of Baumslag–Solitar groups have
$\sigma_{+}(X)=0$ by Example 9.2 and Lemma 9.3.

The above theorems apply to one-relator groups without torsion, but the case
with torsion is also of great interest. The following account reframes the
proof of coherence given in [28] (similar to Wise’s independent proof [53]) in
the language of irreducible curvature.

Consider a one-relator presentation

$G\cong\langle a_{1},\ldots,a_{n}\mid u^{k}\rangle$

where $k>1$. The obvious presentation complex $X$ fails to be concise, and so
$\rho_{+}(X)=1$ by Proposition C, showing that the hypothesis that $G$ should
be torsion-free in Theorem 11.15 cannot be removed. However, there is a
natural modification of the ideas of this paper in which $X$ is allowed to be
an _orbi-complex_ , and the face of $X$ can be endowed with a cone point. In
this modified framework, $G=\pi_{1}(X)$, where the single face of $X$ is
attached along the word $u$ and has a cone point of order $k$. The standard
assignment of areas should give the unique face of $X$ an area of $1/k$.

In this modified framework, the techniques of the proof of Theorem 11.15 apply
as before, except that the areas of faces are divided by $k$.161616Since one-
relator groups are virtually torsion-free, one may also pass to a finite-
sheeted cover which is a genuine complex, as opposed to an orbi-complex, as in
[28, Theorem 2.2]. This avoids the need to discuss orbi-complexes. For any
essential map from an irreducible complex $Y\to X$, the proof of Theorem 11.15
now gives

$k\mathrm{Area}(Y)+\chi(Y_{(1)})\leq 0\,.$

Since the left hand side can be rewritten as $(k-1)\mathrm{Area}(Y)+\tau(Y)$,
it follows that

$\kappa(Y)=\frac{\tau(Y)}{\mathrm{Area}(Y)}\leq 1-k\,.$

In conclusion we have the following estimate on the irreducible curvature of
the presentation orbi-complex of a one-relator group with torsion, which was
implicit in both [28] and [53].

######  Theorem 11.19 (Negative irreducible curvature for one-relator groups
with torsion).

If $X$ is the standard presentation orbi-complex for a one-relator group

$G\cong\langle a_{1},\ldots,a_{n}\mid u^{k}\rangle$

then $\rho_{+}(X)\leq 1-k$. In particular, if $k>1$ then $\rho_{+}(X)<0$.

This is in line with the general expectation that one-relator groups with
torsion are negatively curved. For instance, Newman’s spelling theorem asserts
that all one-relator groups with torsion are hyperbolic [36].

Finally, we make a remark about one-relator groups and Nielsen equivalence.
Propositions 8.2 and 8.3 show that $\rho_{\pm}$ and $\sigma_{\pm}$ are always
invariant under Nielsen equivalence of _presentations_. In the special case of
one-relator groups, this can be extended to Nielsen equivalence of generating
sets by a deep theorem of Magnus.

######  Theorem 11.20 (Magnus’ theorem).

Consider two one-relator presentations

$\langle a_{1},\ldots,a_{m}\mid w\rangle~{}\textrm{and}~{}\langle
a^{\prime}_{1},\ldots,a^{\prime}_{m}\mid w^{\prime}\rangle$

for a group. If $\\{a_{1},\ldots,a_{m}\\}$ and
$\\{a^{\prime}_{1},\ldots,a^{\prime}_{m}\\}$ are Nielsen equivalent as
generating sets then the two presentations are also Nielsen equivalent.

This raises the possibility that our invariants could, at least in principle,
be used to distinguish Nielsen equivalence classes of generating sets of one-
relator groups.

######  Corollary 11.21 (Nielsen equivalence of generating sets for one-
relator groups).

Consider two one-relator presentations

$\langle a_{1},\ldots,a_{m}\mid w\rangle~{}\textrm{and}~{}\langle
a^{\prime}_{1},\ldots,a^{\prime}_{m}\mid w^{\prime}\rangle$

for a group, with corresponding presentation complexes $X$ and $X^{\prime}$
respectively. If $\\{a_{1},\ldots,a_{m}\\}$ and
$\\{a^{\prime}_{1},\ldots,a^{\prime}_{m}\\}$ are Nielsen equivalent as
generating sets then $\rho_{\pm}(X)=\rho_{\pm}(X^{\prime})$ and
$\sigma_{\pm}(X)=\sigma_{\pm}(X^{\prime})$.

###  11.4 Geometric 2-complexes

The next definition allows us to apply these ideas to 3-manifolds.

######  Definition 11.22 (Geometric 2-complex).

A 2-complex $X$ is called _geometric_ if it is endowed with the following
additional structure.

  1. (i)

For each vertex $v$, the link $\mathrm{Lk}(v)$ is equipped with a planar
embedding, i.e. a cellular embedding into $S^{2}$. This determines how to
canonically thicken the 1-skeleton $X_{(1)}$ to a (possibly non-orientable)
handlebody.

  2. (ii)

For each face $f$, the corresponding curve on the surface of the handlebody is
2-sided.

In particular, a presentation is called _geometric_ if its presentation
complex is geometric.

By thickening the 1-skeleton to a handlebody and the faces to 2-handles, a
geometric 2-complex $X$ determines a 3-manifold with boundary $M=M(X)$ that
deformation retracts to $X$. Note that the structure of a geometric complex
pulls back along an essential map.

If $X$ is an irreducible geometric 2-complex then the boundary $\partial M$ of
the thickened 3-manifold is naturally endowed with the structure of a
2-complex, and the inclusion map induces an immersion $\partial M\to X$. The
next fact is morally a consequence of Poincaré duality, which implies that
$\chi(\partial M)=2\chi(M)$ for any compact 3-manifold, although we give a
direct proof using the spherical links of $X$ (cf. [51, Theorem 10.2]).

######  Proposition 11.23 (Poincaré duality).

If $X$ is a finite, irreducible, geometric, branched 2-complex then
$\kappa(\partial M(X))=\kappa(X)$.

###### Proof.

Every face of $X$ appears twice in $\partial M$, so $\mathrm{Area}(\partial
M(X))=2\mathrm{Area}(X)$, and it suffices to prove that $\chi(\partial
M_{(1)})=2\chi(X_{(1)})$. To compute $\chi(\partial M_{(1)})$, we introduce an
auxiliary 2-complex $L$, the disjoint union of the cellular decompositions of
the 2-spheres given by the links of the vertices of $X$.

Since $L$ consists of one sphere for each vertex of $X$, we have
$\chi(L)=2\\#V_{X}$, while $2\\#E_{X}=\\#V_{L}$ by definition. Therefore

$\displaystyle 2\chi(X_{(1)})$ $\displaystyle=$ $\displaystyle
2\\#V_{X}-2\\#E_{X}$ $\displaystyle=$ $\displaystyle\chi(L)-\\#V_{L}$
$\displaystyle=$ $\displaystyle\\#F_{L}-\\#E_{L}\,.$

But $F_{L}$ naturally bijects with $V_{\partial M}$, while $E_{\partial M}$
bijects with $E_{L}$, so $2\chi(X_{(1)})=\chi(\partial M_{(1)})$ as required.
∎

This in turn implies that the irreducible curvature of a 3-manifold spine is
always equal to the surface curvature.

######  Corollary 11.24 (Curvatures of geometric complexes).

If $X$ is a geometric 2-complex then $\rho_{\pm}(X)=\sigma_{\pm}(X)$.

###### Proof.

For any $Y\in\mathrm{Irred}(X)$, the proposition implies that $\kappa(\partial
M(Y))=\kappa(Y)$. Since $\partial M(Y)\to Y\to X$ is an essential map from a
surface, it follows that the range of values attained by maps from irreducible
2-complexes is equal to the range of values attained by maps from surfaces. ∎

Some 2-complexes are not geometric but nevertheless are _virtually_ geometric,
meaning that some finite-sheeted branched cover is geometric [19, Theorem 21].
Virtually geometric 2-complexes were classified by Cashen [12, Theorem 5.9].
Since Corollary 11.24 applies to branched 2-complexes as well as standard
ones, it follows that virtually geometric 2-complexes also satisfy this
conclusion.

######  Corollary 11.25 (Curvatures of virtually geometric 2-complexes).

If $X$ is a virtually geometric 2-complex then
$\rho_{\pm}(X)=\sigma_{\pm}(X)$.

###### Proof.

If $X^{\prime}\to X$ is a geometric branched covering, then
$\rho_{\pm}(X^{\prime})=\sigma_{\pm}(X^{\prime})$ by Corollary 11.24, so

$\rho_{\pm}(X)=\rho_{\pm}(X^{\prime})=\sigma_{\pm}(X^{\prime})=\sigma_{\pm}(X)$

by Proposition 7.4. ∎

Conjecturally, Corollary 11.24 should explain many of the unusually good
properties of 3-manifold groups, such as coherence [42] (cf. Conjecture 12.4
below). In order to apply it, however, one would need bounds on the surface
curvatures of geometric 2-complexes. These bounds are the content of the next
two conjectures. The first is a cousin of Papakyriakopoulos’ sphere theorem
[39].

######  Conjecture 11.26 (Non-positive curvature for geometric complexes).

Let $X$ be a finite, irreducible, geometric 2-complex. If $\sigma_{+}(X)>0$
then either $\partial M(X)$ has a spherical component or $\pi_{1}(X)$ splits
freely.

The analogous conjecture for negative curvature can be thought of as a weak
form of hyperbolisation for Haken 3-manifolds.

######  Conjecture 11.27 (Negative curvature for geometric complexes).

Let $X$ be a finite, irreducible, geometric, branched 2-complex. If
$\sigma_{+}(X)=0$ then either $\partial M(X)$ has a toroidal boundary
component or $\pi_{1}(X)$ has a $\mathbb{Z}^{2}$ subgroup.

###  11.5 Graphs of free groups

Graphs of free groups have been a popular class of examples throughout the
history of geometric group theory. They exhibit a rich range of behaviours and
are very difficult to study in full generality. In the same spirit, I expect
them to be an interesting test class for our rational curvature invariants.
Nothing is currently known, so we will state some conjectures and questions.

Before doing so, however, we should note that there are various possible
choices of 2-complex that one might associate to a given graph of free groups.
Nielsen invariance provides some reassurance, but nevertheless the values of
$\rho_{\pm}$ and $\sigma_{\pm}$ will depend on this choice. Making the correct
choice of 2-complex will necessarily be part of resolving the conjectures and
questions below.

Mapping tori of free group automorphisms and, more generally, endomorphisms,
have proved an especially fruitful class of examples. The following conjecture
posits that they should all be non-positively curved in our strongest sense.

######  Conjecture 11.28 (Irreducible curvatures of mapping tori of free group
endormorphisms).

If $X$ is a compact mapping torus of an injective endomorphism of a finitely
generated free group then $\rho_{+}(X)=0$.

The lower bound $\rho_{+}(X)\geq 0$ follows from the fact that $\chi(X)=0$.
Wise proved that such $X$ have the slightly weaker _non-positive immersions_
property [53, Theorem 6.1]. Adapting his argument may be sufficient to prove
Conjecture 11.28.

The question of which of these examples have negative surface curvature is
likely to be more subtle. The following conjecture is the most optimistic
possible statement.

######  Conjecture 11.29 (Surface curvatures of mapping tori of free group
endormorphisms).

If $X$ is a compact mapping torus of an injective endomorphism of a finitely
generated free group then $\sigma_{+}(X)<0$ unless $\pi_{1}(X)$ has a
Baumslag–Solitar subgroup.

Caution is needed here! Example 11.37 below provides an example of a 2-complex
$X$ with $\sigma_{+}(X)=0$ but $G=\pi_{1}(X)$ is hyperbolic, so in particular
has no Baumslag–Solitar subgroups. Furthermore, $G$ is free-by-cyclic.
However, the 2-complex $X$ is not constructed from the representation of $G$
as a free-by-cyclic group. This example emphasises the importance of choosing
the 2-complex $X$ correctly.

Graphs of free groups with cyclic edge groups have also been a popular source
of examples. I expect the following conjecture to be quite easy to prove, but
it will be a useful warm up for more challenging examples.

######  Conjecture 11.30 (Graphs of free groups with cyclic edge groups).

If $X$ is a compact, connected, irreducible graph of graphs with circular edge
spaces, then $\rho_{+}(X)\leq 0$. Furthermore, $\sigma_{+}(X)<0$ unless
$\pi_{1}(X)$ has a Baumslag–Solitar subgroup.

In the case of general graphs of free groups, it is unclear what the correct
classification should be. However, finding it may shed further light on this
mysterious class.

######  Problem 11.31 (General graphs of free groups).

Determine when a compact, connected graph of graphs $X$ has non-positive or
negative surface or irreducible curvatures.

###  11.6 A botanical map

We finish this section by computing, or at least estimating, the curvatures of
several examples. The intention is to exhibit examples with every combination
of signs of $\rho_{+}$ and $\sigma_{+}$ allowed by Theorem A. The examples are
illustrated in Figure 1.

$\sigma_{+}$$\rho_{+}$$BS(m,n)$Trefoil-knot spineBrady–Crisp groupManning’s
exampleRandom 2-generator1-relator groupRandom 2-generator2-relator
group$F_{2}\times F_{2}$$(1,1)$$S^{2}$$\mathbb{Z}^{4}$Thurston’s
tripus$\mathbb{Z}^{3}$ Figure 1: Some examples of standard, finite,
irreducible 2-complexes $X$ (sometimes referred to by their fundamental
groups) located according to the values $\sigma_{+}(X)$ and $\rho_{+}(X)$. See
Examples 11.32 for $S^{2}$, 11.33 for $\mathbb{Z}^{3}$ and $\mathbb{Z}^{4}$,
11.34 for $F_{2}\times F_{2}$, 11.35 for the trefoil-knot complement, 11.36
for $BS(m,n)$, 11.37 for the Brady–Crisp group, 11.38 for Thurston’s tripus
and 11.39 for Manning’s example. Random 2-generator, 2-relator groups have
$\rho_{+}(X)>0>\sigma_{+}(X)$ by Example 11.12, while random 2-generator,
1–relator groups have $\sigma_{+}(X)<0$ by Example 11.12 and $\rho_{+}(X)=0$
by Corollary 11.16.

We first give an example of a 2-complex at the apex.

######  Example 11.32 (2-sphere).

The standard cellulation $X$ of the 2-sphere $S^{2}$ with two 2-cells has
$\kappa(X)=1$, and so $\sigma_{+}(X)=\rho_{+}(X)=1$ by Example 7.5.

More generally, any 2-complex $X$ at the apex fails to be concise: it either
has two faces with homotopic attaching maps, or it has a face for which the
attaching map is a proper power.

By Proposition C, the region $\sigma_{+}(X)<\rho_{+}(X)=1$ is empty. The
2-skeleta of the $n$-tori provide convenient examples with $\sigma_{+}(X)>0$.

######  Example 11.33 ($n$-torus).

Let $X$ be the standard presentation complex for $\mathbb{Z}^{n}$, which can
also be thought of as the 2-skeleton of the standard cellulation of the
$n$-torus.

If $n=2$ then $\rho_{+}(X)=\sigma_{+}(X)=0$ by Example 7.5.

For $n\geq 3$, we compute

$\kappa(X)=\frac{1-n+\binom{n}{2}}{\binom{n}{2}}=1-2/n$

so $1>\rho_{+}(X)\geq 1-2/n$.171717I expect this last estimate to be an
equality; it should be possible to check this using Wise’s sectional curvature
for angle structures [51].

To compute $\sigma_{+}(X)$, on the one hand Corollary 11.8 gives that
$\sigma_{+}(X)\leq 1/3$ since every face of $X$ is a square and the link of
the unique vertex has girth 3. On the other hand, since $n\geq 3$ there is an
obvious immersion $C\to X$, where $C$ is the 2-skeleton of a cube, and so
$\sigma_{+}(X)\geq\kappa(C)=1/3$. In summary, we have

$\sigma_{+}(X)=1/3\leq 1-2/n\leq\rho_{+}(X)<1\,.$

If $n=3$, $X$ is geometric and so we have $\rho_{+}(X)=\sigma_{+}(X)=1/3$,
while for $n>3$ we have $\rho_{+}(X)>\sigma_{+}(X)$.

Thus, $n$-tori provide a family of examples in the region
$0<\sigma_{+}(X)\leq\rho_{+}(X)<1$. Moving downwards in Figure 1, the direct
product of two free groups provides an easy example of a 2-complex with
$0=\sigma_{+}(X)<\rho_{+}(X)$.

######  Example 11.34 ($F_{2}\times F_{2}$).

The standard complex $X$ associated to the presentation

$\langle a_{1},a_{2},b_{1},b_{2}\mid[a_{i},b_{j}]=1\,,\,i,j=1,2\rangle$

of the group $F_{2}\times F_{2}$ is a locally $CAT(0)$ Euclidean complex, so
has $\sigma_{+}(X)=0$. Indeed, Corollary 11.6 implies that $\sigma_{+}(X)\leq
0$, while the obvious geodesic inclusion of a torus implies that
$\sigma_{+}(X)\geq 0$ by monotonicity. On the other hand,

$\rho_{+}(X)\geq\frac{\tau(X)}{\mathrm{Area}(X)}=1/4>0\,.$

Likewise, there are many examples with $\sigma_{+}(X)<0<\rho_{+}(X)$. For
instance, we saw in Example 11.12 that the standard complex associated to a
randomly chosen 2-generator, 2-relator presentation has this property.

We now turn our attention to examples with $\rho_{+}(X)\leq 0$. The origin
$\sigma_{+}(X)=\rho_{+}(X)=0$ exhibits an especially rich array of behaviours.
The 2-torus is the most obvious example, but we list three more illustrative
examples here.

######  Example 11.35 (Trefoil knot).

Let $X$ be the standard presentation complex of

$\langle a,b\mid a^{2}b^{-3}\rangle\,,$

the fundamental group of the trefoil-knot complement. As a one-relator
presentation complex it has $\rho_{+}(X)\leq 0$ by Theorem 11.15, but
$\kappa(X)=0$ so $\rho_{+}(X)=0$. Since $X$ is geometric,
$\sigma_{+}(X)=\rho_{+}(X)=0$ by Corollary 11.24.

We already saw that presentation complexes of Baumslag–Solitar groups have
$\rho_{+}(X)=0$. In fact, the same is true of their maximal surface curvature.

######  Example 11.36 (Baumslag–Solitar groups revisited).

Let $X$ be the presentation complex of

$\langle a,b\mid ba^{m}b^{-1}a^{-n}\rangle\,,$

the standard presentation of the $(m,n)$ Baumslag–Solitar group. We saw that
$\rho_{+}(X)=0$ in Example 11.17. But [19, Theorem 21] asserts that $X$ is
virtually geometric, so $\sigma_{+}(X)=0$ by Corollary 11.25.

The next example is less well known, but is important because it provides an
example with $\rho_{+}(X)=\sigma_{+}(X)=0$ and with hyperbolic fundamental
group. This example was studied by Brady and Crisp [4], and we name it for
them,

######  Example 11.37 (Brady–Crisp group).

Let $X$ be the presentation complex of the _Brady–Crisp group_

$G=\langle a,b\mid aba^{2}b^{-2}\rangle\,.$

As a 2-generator, one-relator presentation, we have $\rho_{+}(X)=0$ by
Corollary 11.16. The link $L$ of the unique vertex of $X$ is a tetrahedron,
which is in particular a 3-valent planar graph. But any such 2-complex is
virtually geometric, via an argument similar to [19, §6]. We sketch the
argument here.

Because $L$ is planar, we can think of it as embedded on the surface of a
3-ball $B$. We now thicken each vertex $v$ of $L$ to a disc $D_{v}$ on
$\partial B$. Each vertex of $L$ has an opposite vertex ${v}^{*}$. The edges
of $L$ incident at $v$ are identified with the edges of $L$ incident at
${v}^{*}$, and since $L$ is trivalent we can extend this identification to a
homeomorphism $D_{v}\cong D_{{v}^{*}}$. Gluing up the discs in pairs using
these homeomorphisms constructs a (possibly non-orientable) handlebody $U$,
which is naturally a thickening of the 1-skeleton $X_{(1)}$, and by
construction the attaching curves of the faces are embedded on the boundary
$\partial U$. This does not show that $X$ is geometric, because the curves may
not be 2-sided. However, passing to an orientable double cover $U^{\prime}\to
U$ and pulling back the attaching maps of the faces constructs a degree-two
branched cover $X^{\prime}\to X$ which is certainly geometric, because every
embedded curve on an orientable surface is 2-sided.

In summary, $X$ is virtually geometric, so $\sigma_{+}(X)=\rho_{+}(X)=0$ by
Corollary 11.25.

The example of the Brady–Crisp group emphasises that, in some sense, our
invariants only see the 2-dimensional features of a 2-complex. Indeed, Brady
and Crisp studied $G$ as an example of a hyperbolic group with CAT(0)
dimension $2$ but CAT(-1) dimension $3$. If it were possible to put a
polyhedral CAT(-1) structure on $X$ then we would have $\sigma_{+}(X)<0$, by
Corollary 11.7.

Any $C(7)$ two-generator, one-relator presentation (for instance, a randomly
chosen one will do) provides an example with $\rho_{+}(X)=0$ but
$\sigma_{+}(X)<0$.

Finally, we give examples with $\rho_{+}(X)<0$. Corollary 11.24 suggests the
spine of a hyperbolic 3-manifold with totally geodesic boundary. Thurston’s
tripus provides a famous example.

######  Example 11.38 (Thurston’s tripus).

In [45, §3.2], Thurston constructed a compact hyperbolic 3-manifold $T$ with
boundary a totally geodesic surface of genus 2, by identifying faces of a
hyperbolic polyhedron. The manifold $T$ is sometimes called the _tripus_.
Paoluzzi–Zimmerman computed a presentation of the fundamental group as

$\pi_{1}(T)=\langle a,b,c\mid
ab^{-1}a^{-1}bc^{-1}b^{-1}ca^{-1}c^{-1}\rangle\,,$

and indeed this presentation can be seen as dual to the polyhedron
description, which means that the presentation complex $X$ is geometric and
$M(X)\cong T$ [38].

The presentation complex $X$ provides an example with
$\rho_{+}(X)=\sigma_{+}(X)<0$. To see this, note that the link of the unique
vertex of $X$ is the 1-skeleton of a triangular prism, and in particular has
girth equal to 3. The face has length 9 and so

$\sigma_{+}(X)\leq 1-\frac{9}{2}\left(1-\frac{2}{3}\right)=-\frac{1}{2}$

by Corollary 11.8. Meanwhile, $\rho_{+}(X)=\sigma_{+}(X)$ since $X$ is
geometric.

Finally, we finish with an example satisfying $\sigma_{+}(X)<\rho_{+}(X)<0$.
This example was studied by Manning [34], who exhibited it as the first
example of a one-relator presentation complex that is not virtually geometric.
The invariants $\rho_{+}$ and $\sigma_{+}$ enable us to recover Manning’s
result by other means.

######  Example 11.39 (Manning’s example).

Let $X$ be the standard complex associated to the presentation

$\langle a,b,c\mid b^{2}a^{2}c^{2}abc\rangle\,.$

The link of the vertex is the complete bipartite graph $K_{3,3}$ [34, Example
2.1], which has girth 4, so

$\sigma_{+}(X)\leq 1-\frac{9}{2}\left(1-\frac{2}{4}\right)=-5/4$

by Corollary 11.8. On the other hand, $\rho_{+}(X)\geq\chi(X)=-1$. Therefore,
$X$ provides an example with

$\sigma_{+}(X)<\rho_{+}(X)\,.$

By Corollary 11.25, it follows that $X$ is not virtually geometric, which
gives an alternative proof of [34, Corollary 4.2]

To prove that $\rho_{+}(X)<0$ requires a more delicate argument. One approach
is to compute the primitivity rank,

$\pi(b^{2}a^{2}c^{2}abc)=3\,.$

Theorem 10.7 now implies that every $Y\in\mathrm{Irred}(X)$ has
$\tau(Y)\leq-1$, so they also have $\kappa(Y)<0$, and therefore
$\rho_{+}(X)<0$ by the rationality theorem.

Alternatively, for an effective estimate, one can put an angle structure on
$X$ and adapt Wise’s generalised sectional curvature to the setting of
branched immersions, as suggested in Remark 11.9. In fact, setting the length
of every edge in the link equal to $5\pi/9$ leads to the estimate
$\rho_{+}(X)\leq-1$, and since $\chi(X)=-1$ we actually have equality.

##  12 Properties of groups with curvature bounds

In this section, we collect known consequences of curvature bounds on
2-complexes, each of which can be considered as a contribution to the
geography problem. The currently known consequences all first appeared in work
on one-relator groups. We also make some further conjectures. The conjectural
picture is summarised in a geographical map – see Figure 2.

###  12.1 Concise 2-complexes

We start by noting the proof of Proposition C, which classifies the
2-complexes with $\rho_{+}(X)=1$.

###### Proof of Proposition C.

Suppose that $\rho_{+}(X)=1$. First, we show that $X$ fails to be concise.

By the rationality theorem, $\rho_{+}(X)$ is realised by an essential map from
an irreducible 2-complex $Y$ with $\kappa(Y)=1$, so $\chi(Y_{(1)})=0$. Without
loss of generality $Y$ is connected, so $Y_{(1)}$ is a circle; write
$\pi_{1}(Y_{(1)})=\langle w\rangle$. Therefore, the attaching maps of the
faces of $Y$ represent non-zero powers of $w$ in the fundamental group. If $Y$
were concise then there could be only one such face, and it would be
represented by $w$, but this would contradict the fact that $Y$ is
irreducible. So $X$ is not concise.

Second, note that $Y$ is an irrigid complex in the sense of Defiinition 9.1
and so, by Proposition 9.3, there is an essential map from a surface $S\to Y$
with $\kappa(Y)=1$, which in turn implies that $\sigma_{+}(X)=1$ ∎

$\sigma_{+}$$\rho_{+}$Aspherical,solvable word
problem,coherentHyperbolic,locally
quasiconvexHyperbolic,coherentHyperbolicAspherical,solvableword problem$(1,1)$
Figure 2: Conjectural properties of a finite, irreducible 2-complex $X$ and
its fundamental group $G=\pi_{1}(X)$, depending on the invariants
$\sigma_{+}(X)$ and $\rho_{+}(X)$. The vertex $\sigma_{+}(X)=\rho_{+}(X)=1$
consists of 2-complexes with either two equal faces or a single face for which
the attaching map is a proper power. The dashed line
$\sigma_{+}(X)<\rho_{+}(X)=1$ is empty.

###  12.2 Non-positive irreducible curvature

Recall that a map of 2-complexes $Y\to X$ is an _immersion_ if it is locally
injective everywhere. In 2003, Wise introduced the following definition [50].

######  Definition 12.1 (Non-positive immersions).

A finite, standard 2-complex $X$ has _non-positive immersions_ if, for any
immersion from a finite, connected, standard 2-complex $Y\to X$, either
$\chi(Y)\leq 0$ or $Y$ is simply connected. If $\chi(Y)>0$ implies that $Y$ is
contractible, then $X$ is said to have _contracting non-positive immersions_.

The ideas around this definition have been developed by Wise more recently in
[53] and [52].

It is not hard to see that the non-positive-immersions property is a slight
weakening of our notion of non-positive irreducible curvature.

######  Lemma 12.2 (Non-positive irreducible curvature implies non-positive
immersions).

Let $X$ be a finite, standard 2-complex. If $\rho_{+}(X)\leq 0$ then $X$ has
contracting non-positive immersions.

###### Proof.

For any immersion from a finite, connected complex $Y\to X$, consider an
unfolding

$Y^{\prime}=G\vee Y_{1}\vee\ldots\vee Y_{m}\vee D_{1}\vee\ldots\vee D_{n}$

as in Lemma 4.6. Since $Y$ and $Y^{\prime}$ are homotopy equivalent, the Euler
characteristic of $Y$ can be written as

$\chi(Y)=\chi(G)+\sum_{i=1}^{m}(\chi(Y_{i})-1)\,.$

The graph $G$ is connected so $\chi(G)\leq 1$, and the hypothesis on
$\rho_{+}(Y_{i})$ implies that

$\chi(Y_{i})=\tau(Y_{i})\leq 0\,,$

so $\chi(Y)\leq 1$ with equality only if $G$ is a tree and $m=0$. Therefore,
$\chi(Y)>0$ implies that $Y$ is homotopy equivalent to a tree wedged with
discs, and is in particular contractible, as required. ∎

It is an open question whether there is an example of a 2-complex $X$ with
contracting non-positive immersions but $\rho_{+}(X)>0$. In any case, I
believe that the class of 2-complexes with non-positive irreducible curvature
may be more useful than the possibly larger class of 2-complexes with non-
positive immersions. Indeed, the fact that non-positive irreducible curvature
can be recognised algorithmically is one argument in favour of this point of
view.

In any case, Lemma 12.2 enables us to exploit the known properties of
2-complexes with non-positive immersions that Wise has established. The
following theorem, which contributes to the geography problem, follows from
[53, Theorems 1.3, 1.6] using Lemma 12.2. Recall that a group $G$ is said to
be _locally indicable_ if every non-trivial, finitely generated subgroup $H$
of $G$ surjects $\mathbb{Z}$.

######  Theorem 12.3 (Geography with non-positive irreducible curvature).

Let $X$ be a finite, standard, irreducible 2-complex. If $\rho_{+}(X)\leq 0$
then:

  1. (i)

$X$ is aspherical, and

  2. (ii)

$\pi_{1}(X)$ is locally indicable.

The proofs in [53] use tower arguments, a technique going back to
Papakyriakopoulos’ proof of Dehn’s lemma [39] and also used by Howie in his
locally indicable Freiheitssatz [22]. The folding technique of Lemma 2.5 can
also be used to give a direct proof. Combined with Theorem 11.15, the theorem
recovers the Cockroft–Lyndon asphericity theorem for torsion-free one-relator
groups [31, 15], as well as Brodskiĭ–Howie’s theorem that torsion-free one-
relator groups are locally indicable [7, 23].

A group $G$ is _coherent_ if every finitely generated subgroup is finitely
presented. A notorious question of Baumslag asks whether every one-relator
group is coherent. Motivated by Baumslag’s question, Wise conjectured that the
fundamental group of any finite 2-complex $X$ with non-positive immersions is
coherent [53, Conjecture 1.10]. The following conjecture sharpens Wise’s
conjecture slightly.

######  Conjecture 12.4 (Non-positive irreducible curvature implies
coherence).

Let $X$ be a finite, standard, irreducible 2-complex. If $\rho_{+}(X)\leq 0$
then $\pi_{1}(X)$ is coherent.

We shall see below that this conjecture is true when $\rho_{+}(X)<0$.

###  12.3 Negative irreducible curvature

Negative irreducible curvature puts even stronger constraints on a 2-complex
$X$. Recently, Louder and the author strengthened Wise’s definition of non-
positive immersions to _negative immersions_ [30, Definition 1.1], and then
further to _uniform negative immersions_ [29, Definition 3.23].

######  Definition 12.5.

A finite, standard 2-complex $X$ has _negative immersions_ if, for every
immersion from a finite, connected complex $Y\to X$, either $\chi(Y)<0$ or $Y$
is reducible. If, furthermore, there is an $\epsilon>0$ such that

$\kappa(Y)=\frac{\chi(Y)}{\mathrm{Area}(Y)}\leq-\epsilon$

for every such immersion $Y\to X$, then $X$ has _uniform negative immersions_.

######  Remark 12.6.

Wise makes use of a different definition of negative immersions for
2-complexes in [53]. See [29, §3.4] for a comparison between Definition 12.5
and Wise’s definition.

Negative irreducible curvature is clearly stronger than uniform negative
immersions, since the definitions are the same except that the class of maps
considered expands from immersions to branched immersions. The next lemma is
therefore immediate.

######  Lemma 12.7 (Negative irreducible curvature implies uniform negative
immersions).

If $X$ is a finite, standard, 2-complex and $\rho_{+}(X)<0$ then $X$ has
uniform negative immersions.

The following theorem, which contributes to the geography problem for
complexes of negative irreducible curvature, lists some very strong
constraints on the finitely generated subgroups of the fundamental groups of
complexes with negative irreducible curvature.

######  Theorem 12.8 (Geography with negative irreducible curvature).

If $X$ is a finite, irreducible, standard 2-complex and $\rho_{+}(X)<0$, then
the fundamental group $G$ satisfies the following conditions:

  1. (i)

$G$ is 2-free (i.e. every 2-generator subgroup is free);

  2. (ii)

$G$ is coherent (i.e. every finitely generated subgroup is finitely
presented);

  3. (iii)

each finitely generated, one-ended subgroup $H$ of $G$ is co-Hopfian (i.e.
every injective self-homomorphism of $H$ is an automorphism);

  4. (iv)

for each integer $r$, there are only finitely many conjugacy classes of
finitely generated, one-ended subgroups $H$ of $G$ such that the
abelianisation of $H$ has rational rank at most $r$;

  5. (v)

each finitely generated subgroup $H$ of $G$ is either cyclic or large in the
sense of Pride (i.e. some subgroup $H_{0}$ of finite index in $H$ surjects a
non-abelian free group);

###### Proof.

Items (ii), (iii), (iv) and (v) are the immediate results of combining Lemma
12.7 with [29, Corollary 6.4 and Theorem 6.5].

Item (i) is implicit in [30], but we give an explicit proof here. Suppose that
$H$ is a 2-generator subgroup of $G$, which we may assume is freely
indecomposable by Grushko’s theorem. Now $H$ is represented up to conjugacy by
an immersion $Y\to X$, where $Y$ is a finite, connected, irreducible
2-complex, by [29, Theorem 6.3]. The hypothesis that $\rho_{+}(X)<0$ implies
that

$\tau(Y)=\chi(Y)=1-b_{1}(Y)+b_{2}(Y)<0$

so, since $b_{1}(Y)\leq 2$, it follows that both $b_{1}(Y)=2$ and
$b_{2}(Y)=0$. Finally, a theorem of Stallings implies that $H$ is free, as
required [43]. ∎

The combination of Theorems 11.18 and 12.8 recover the properties of one-
relator groups with negative immersions discovered in [30] and [29].

Theorems 12.3 and 12.8(i) together imply that if $\rho_{+}(X)<0$ then
$G=\pi_{1}(X)$ is a group of finite type without Baumslag–Solitar subgroups,
which naturally raises the question of whether $G$ is hyperbolic. If so, then
one might further hope that the some of the constraints on the subgroups of
$G$ implied by Theorem 12.8 come from the geometry of those subgroups. These
considerations motivate the following conjecture, which sharpens [30,
Conjecture 1.9] and [53, Conjecture 5.1].

######  Conjecture 12.9 (Negative irreducible curvature implies locally
quasiconvex).

Let $X$ be a finite, standard, irreducible 2-complex. If $\rho_{+}(X)<0$ then
$\pi_{1}(X)$ is a locally quasiconvex hyperbolic group.

This should be compared to Conjecture 12.12 below, which relaxes the
hypotheses to $\sigma_{+}(X)<0$. Italiano, Martelli and Migliorini recently
gave the first example of a group of finite type without Baumslag–Solitar
subgroups [24], but no 2-dimensional examples are known. A recent theorem of
Linton [26] proves that one-relator groups with negative immersions – i.e.
with $\rho_{+}(X)<0$ – are hyperbolic, confirming part of Conjecture 12.9 in
the one-relator case.

###  12.4 Surface curvature bounds

As we have seen in the previous two sections, recent developments in the
theory of one-relator groups can be seen as explorations of the consequences
of upper bounds on irreducible curvature. Surface curvature has not yet been
as thoroughly investigated, and so the constraints that surface curvature
bounds impose on $X$ are still conjectural. We list some of those conjectures
here. We have already seen many examples of complexes with upper bounds on
their surface curvature, and these conjectures are consistent with those
examples. Throughout, $X$ is a finite, standard, irreducible 2-complex.

Non-positive curvature is frequently used in geometric group theory to prove
that a space is aspherical: this is the Cartan–Hadamard theorem for CAT(0)
spaces [6, Theorem II.4.1], a standard result about small-cancellation
complexes [14], and the Lyndon asphericity theorem for one-relator groups [31,
15]. We have already seen that $X$ is aspherical if it has non-positive
irreducible curvature, but we conjecture that this result extends to the much
wider context of non-positive surface curvature.

######  Conjecture 12.10 (Non-positive surface curvature implies aspherical).

If $\sigma_{+}(X)\leq 0$ then $X$ is aspherical.

A suitable formulation of van Kampen’s lemma implies that any map from a
2-sphere to $X$ is homotopic to a morphism of 2-complexes $S\to X$, where the
realisation of $S$ is a 2-sphere. The challenge of Conjecture 12.10 is to
improve this morphism to an essential map.

Another frequent use of non-positive curvature in group theory is to the solve
the word problem: this follows from the quadratic isoperimetric inequality for
CAT(0) groups [6, Proposition III.$\Gamma$.1.6], from Greendlinger’s lemma for
small-cancellation groups [32], and is a famous theorem of Magnus for one-
relator groups [33]. Our second conjecture proposes that, in the case of
2-complexes, all of these can be deduced from a bound on surface curvature.

######  Conjecture 12.11 (Non-positive surface curvature implies solvable word
problem).

If $\sigma_{+}(X)\leq 0$ then the word problem is solvable in $\pi_{1}(X)$.

The last two conjectures suggest that non-positive surface curvature is
similar to other notions of non-positive curvature commonly used in geometric
group theory. However, as noted after Theorem 11.15, there are 2-complexes of
non-positive surface curvature with very large isoperimetric inequality, and
in particular $\sigma_{+}(X)$ does not imply any of the standard notions of
non-positive curvature already used in geometric group theory.

However, it is likely that negative irreducible curvature is closely related
to the dominant notion of negative curvature for groups, namely word-
hyperbolicity.

######  Conjecture 12.12 (Negative surface curvature implies hyperbolic).

If $\sigma_{+}(X)<0$ then $\pi_{1}(X)$ is word-hyperbolic.

This conjecture posits a common generalisation of Greendlinger’s lemma for
small-cancellation groups and the B.B. Newman spelling theorem for one-relator
groups with torsion. It should also shed light on the cases of non-positively
curved Euclidean 2-complexes and one-relator presentation complexes, where
there is not yet a complete understanding of which groups are word-hyperbolic.
However, the Example 11.37 provides an example of a non-positively curved,
Euclidean, one-relator 2-complex, for which hyperbolicity of the fundamental
group is not a consequence of Conjecture 12.12.

The conjectures to this point all concern upper curvature bounds. However, it
is possible that lower bounds may also provide useful information.

######  Conjecture 12.13 (Positive surface curvature implies no surface
subgroups).

If $\sigma_{-}(X)>0$ then no subgroup of $\pi_{1}(X)$ is isomorphic to the
fundamental group of a closed surface $S$ with $\chi(S)\leq 0$.

The challenge of Conjecture 12.13 is to prove that _any_ finite 2-complex $X$
with $\pi_{1}(X)$ a surface group has $\rho_{-}(X)\leq 0$.

In §9, we saw that Baumslag–Solitar groups are closely related to surfaces,
because both can be thought of as the fundamental groups of irrigid complexes.
An even more intriguing possibility is that $\sigma_{-}(X)>0$ also implies
that $\pi_{1}(X)$ has no Baumslag–Solitar subgroups. There are very few known
examples of infinite, finitely presented groups without surface or
Baumslag–Solitar subgroups.

## References

  * [1] Benjamin Baumslag and Stephen J. Pride.  Groups with two more generators than relators.  Journal of the London Mathematical Society. Second Series, 17(3):425–426, 1978.
  * [2] Gilbert Baumslag.  A non-cyclic one-relator group all of whose finite quotients are cyclic.  J. Austral. Math. Soc., 10:497–498, 1969.
  * [3] Gilbert Baumslag.  Some problems on one-relator groups.  In Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), volume 372 of Lecture Notes in Math., pages 75—-81, Berlin, 1974. Springer.
  * [4] Noel Brady and John Crisp.  $\rm CAT(0)$ and $\rm CAT(-1)$ dimensions of torsion free hyperbolic groups.  Comment. Math. Helv., 82(1):61–85, 2007.
  * [5] Martin R. Bridson.  The geometry of the word problem.  In Invitations to geometry and topology, volume 7 of Oxf. Grad. Texts Math., pages 29–91. Oxford Univ. Press, Oxford, 2002.
  * [6] Martin R. Bridson and André Haefliger.  Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [7] S. D. Brodskiĭ.  Equations over groups and groups with one defining relation.  Uspekhi Mat. Nauk, 35(4(214)):183, 1980.
  * [8] Danny Calegari.  Surface subgroups from homology.  Geometry & Topology, 12(4):1995–2007, 2008.
  * [9] Danny Calegari.  scl.  Number 20 in MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009\.