# From letter-quasimorphisms to angle structures and spectral gaps for scl

Alexis Marchand  DPMMS, Centre for Mathematical Sciences, Wilberforce Road,
Cambridge CB3 0WB, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We give a new geometric proof of a theorem of Heuer showing that, in the
presence of letter-quasimorphisms (which are analogues of real-valued
quasimorphisms with image in free groups), and in particular in RAAGs, there
is a sharp lower bound of $\frac{1}{2}$ for stable commutator length. Our
approach is to show that letter-quasimorphisms give rise to negatively curved
angle structures on admissible surfaces. This generalises Duncan and Howie’s
proof of the $\frac{1}{2}$-lower bound in free groups, and can also be seen as
a version of Bavard Duality for letter-quasimorphisms.

##  1\. Introduction

Stable commutator length, or $\operatorname{scl}$, is an invariant of group
elements that measures their homological complexity. Topologically, given a
group $G$ realised as the fundamental group of a space $X$, and given a
homologically trivial element $g\in G$, the _stable commutator length_ of $g$
is defined as

$\operatorname{scl}(g)=\inf_{f,\Sigma}\frac{-\chi^{-}(\Sigma)}{2n(\Sigma)},$

where the infimum is over all _admissible surfaces_ , i.e. maps
$f:\left(\Sigma,\partial\Sigma\right)\rightarrow\left(X,g\right)$ from
oriented compact surfaces, bounding $g^{n(\Sigma)}$ for some
$n(\Sigma)\in\mathbb{N}$ — see §2 for a precise definition.

Dually, Bavard [bavard] showed that $\operatorname{scl}$ can be interpreted in
terms of quasimorphisms:

$\operatorname{scl}(g)=\sup_{\phi}\frac{\phi(g)}{2D(\phi)},$

where the supremum is over all maps $\phi:G\rightarrow\mathbb{R}$ which
satisfy

$D(\phi)=\sup_{a,b\in G}\left|\phi(ab)-\phi(a)-\phi(b)\right|<\infty,$

and $\phi\left(a^{n}\right)=n\phi(a)$ for all $a\in G$ and $n\in\mathbb{Z}$ —
such maps are called _homogeneous quasimorphisms_. See Calegari’s book [cal-
scl]*§2.5 for more on Bavard Duality.

This paper is about proving _lower bounds_ for $\operatorname{scl}$. A guiding
thread is that $\operatorname{scl}$ takes large values in the presence of
negative curvature. So-called spectral gaps (defined in §2.b) have been
discovered in groups with various notions of hyperbolicity or non-positive
curvature, including Gromov-hyperbolic groups [calegari-fujiwara], mapping
class groups [bbf], $3$-manifold groups [chen-heuer], and certain amalgamated
free products [cfl, chen-specgaps]. Of particular importance is the following:

######  Theorem 1.1 (Duncan–Howie [duncan-howie]).

Suppose that there is a morphism $\varphi:G\rightarrow F$, with $F$ a free
group, such that $\varphi(g)\neq 1$. Then
$\operatorname{scl}(g)\geq\frac{1}{2}$.

In particular, (residually) free groups satisfy a strong spectral gap for
$\operatorname{scl}$.

_Strong spectral gaps_ are sharp spectral gaps at $\frac{1}{2}$ with an
additional assumption of non-vanishing of $\operatorname{scl}$ outside the
trivial element (see Definition 2.2). Strong spectral gaps are inherited by
subgroups, so they could reveal information about the subgroup structure of
certain groups.

The Duncan–Howie Theorem was generalised by Heuer [heuer], who showed that
right-angled Artin groups (RAAGs) also satisfy a strong spectral gap for
$\operatorname{scl}$. A remarkable corollary is that the fundamental group of
any special cube complex has a strong spectral gap. Other authors had
previously studied spectral gaps in RAAGs, including Fernós et al. [fft], and
Forester et al. [fst].

In order to obtain a strong spectral gap for RAAGs, Heuer [heuer] introduced
_letter-quasimorphisms_ , which are analogues of quasimorphisms with image in
free groups (see Definition 3.1). We will say that a group is _quasi-
residually free_ if every element admits a letter-quasimorphism with
sufficiently non-trivial image (see Definition 3.3); examples include
residually free groups and RAAGs. Heuer’s main result is the following:

######  Theorem 1.2 (Heuer [heuer]).

Suppose that there is a letter-quasimorphism $\Phi:G\rightarrow\mathcal{A}$
with $\Phi(g)\neq 1$ and $\Phi\left(g^{n}\right)=\Phi(g)^{n}$ for all
$n\in\mathbb{Z}$. Then $\operatorname{scl}(g)\geq\frac{1}{2}$.

In particular, quasi-residually free groups satisfy a strong spectral gap for
$\operatorname{scl}$.

Heuer’s Theorem can be seen as a coarse version of the Duncan–Howie Theorem;
it can also be thought of as a kind of Bavard Duality for letter-
quasimorphisms.

The main goal of this paper is to obtain Theorem 1.2 from a geometric route
that makes apparent the role played by negative curvature. Heuer’s strategy of
proof [heuer] is to use letter-quasimorphisms to construct real-valued
quasimorphisms and apply classical Bavard Duality. Our approach is to show
that, in the presence of letter-quasimorphisms, any admissible surface admits
a cellular decomposition giving rise to a certain negatively curved geometric
structure. This geometric structure is an _angle structure_ in the sense of
Wise [wise:ang] — see §4 below for a definition. We give an upper bound on the
curvature of admissible surfaces for this angle structure, which yields an
estimate of their Euler characteristic via the Combinatorial Gauß–Bonnet
Formula. This approach can be seen as a geometric reinterpretation of Chen’s
linear programming duality method [chen-specgaps, chen-heuer, chen:kervaire].

This is the content of our main theorem:

######  Theorem A.

Let $X$ be a connected $2$-complex and let $g\in\pi_{1}X\smallsetminus 1$.
Assume that there is a letter-quasimorphism
$\Phi:\pi_{1}X\rightarrow\mathcal{A}$ with $\Phi(g)\neq 1$ and
$\Phi\left(g^{n}\right)=\Phi(g)^{n}$ for all $n\in\mathbb{Z}$.

Then given a monotone, incompressible, disc- and sphere-free admissible
surface $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ with
$f_{*}\left[\partial\Sigma\right]=n(\Sigma)\left[S^{1}\right]$ for some
$n(\Sigma)\in\mathbb{N}_{\geq 1}$, there is an angle structure on $\Sigma$
whose total curvature satisfies

$\kappa\left(\Sigma\right)\leq-2\pi\cdot n(\Sigma).$

In particular, $\operatorname{scl}(g)\geq\frac{1}{2}$.

There are several advantages to our geometric route. One of them is that it
emphasizes the kinship of Heuer’s Theorem with the Duncan–Howie Theorem on the
one hand, and the Bavard Duality Theorem on the other hand. Indeed, Duncan and
Howie’s proof [duncan-howie] uses a language that is very close to Wise’s
angle structures, and our argument is of a similar flavour. Likewise, Bavard’s
work [bavard] shows how the data of a quasimorphism relates to topological
information on admissible surfaces, and our approach generalises this to
letter-quasimorphisms. Hence, our proof brings together three of the most
important theorems to date on lower bounds for $\operatorname{scl}$. The hope
is that, by bringing negative curvature to the forefront, our method could
lead to further spectral gap results in other hyperbolic-like settings.

### Outline of the paper

We start by recalling the topological definition of $\operatorname{scl}$ and a
few basic facts about the reduction of admissible surfaces in §2. We then go
on to define letter-quasimorphisms in §3, following Heuer [heuer]. In §4, we
introduce angle structures, following Wise [wise:ang]. The core of the paper
is §5, in which we explain how letter-quasimorphisms give rise to negatively
curved angle structures on admissible surfaces, leading to Theorem A.

### Acknowledgements

As always, my gratitude goes to Henry Wilton for his constant support and
enthusiasm, and for the always helpful mathematical discussions. This work was
financially supported by an EPSRC PhD studentship.

##  2\. Stable commutator length

###  2.a. Topological definition

We briefly recall the topological definition of $\operatorname{scl}$ — for a
detailed treatment, we refer the reader to Calegari’s monograph [cal-scl].

Let $G$ be a group, and let $X$ be a connected topological space with
$\pi_{1}X=G$. Pick an element $g\in G$, whose conjugacy class corresponds to
the free homotopy class of a loop $\gamma:S^{1}\rightarrow X$. Note that the
circle $S^{1}$ is oriented: we have a well-defined fundamental class
$\left[S^{1}\right]\in H_{1}\left(S^{1}\right)$. An _admissible surface_ for
$g$ in $X$ is the data of an oriented compact surface $\Sigma$, and of
continuous maps $f:\Sigma\rightarrow X$ and $\partial
f:\partial\Sigma\rightarrow S^{1}$, such that the following diagram commutes:

$\partial\Sigma$$\Sigma$$S^{1}$$X$$\iota$$\gamma$$\partial f$$f$

where $\iota:\partial\Sigma\hookrightarrow\Sigma$ is the inclusion. Such an
admissible surface will be denoted by
$f:\left(\Sigma,\partial\Sigma\right)\rightarrow\left(X,g\right)$.

The complexity of a connected surface $\Sigma$ is measured by its _reduced
Euler characteristic_
$\chi^{-}\left(\Sigma\right)=\min\left\\{0,\chi(\Sigma)\right\\}$. If $\Sigma$
is disconnected, we set $\chi^{-}(\Sigma)=\sum_{K}\chi^{-}(K)$, where the sum
is over all connected components of $\Sigma$.

######  Definition 2.1.

The _stable commutator length_ of an element $g\in G$ is

$\operatorname{scl}(g)=\inf_{f,\Sigma}\frac{-\chi^{-}(\Sigma)}{2n(\Sigma)},$

where the infimum is over all admissible surfaces
$f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ with $\partial
f_{*}\left[\partial\Sigma\right]=n(\Sigma)\left[S^{1}\right]$ in
$H_{1}\left(S^{1}\right)$, for some $n(\Sigma)\in\mathbb{N}_{\geq 1}$. We
agree that $\inf\varnothing=\infty$.

###  2.b. Spectral gaps

This paper is concerned with lower bounds for $\operatorname{scl}$. More
precisely, we will say that a group $G$ has a _spectral gap_ if there is a
constant $\varepsilon>0$ such that any $g\in G$ with $\operatorname{scl}(g)>0$
has $\operatorname{scl}(g)\geq\varepsilon$.

It is well-known that in any group $G$, a non-trivial commutator always has
$\operatorname{scl}\leq\frac{1}{2}$ (because a commutator is always bounded by
a torus with one boundary component). Hence, the best spectral gap that can
occur is $\varepsilon=\frac{1}{2}$, which motivates the following definition:

######  Definition 2.2.

A group $G$ is said to have a _strong spectral gap_ if

$\forall g\in G\smallsetminus 1,\>\operatorname{scl}(g)\geq\frac{1}{2}.$

It is notable that the property of having a strong spectral gap descends to
subgroups by monotonicity of $\operatorname{scl}$.

Groups that are known to have a strong spectral gap include residually free
groups by work of Duncan and Howie [duncan-howie]. Heuer [heuer] gave a far-
ranging generalisation of the Duncan–Howie Theorem, proving that groups with
enough letter-quasimorphisms (including all right-angled Artin groups and
their subgroups) have a strong spectral gap. The goal of this paper is to give
an alternative account of Heuer’s Theorem, with a more topological proof that
is closer in spirit to Duncan and Howie’s.

###  2.c. Reduction of admissible surfaces

It is standard to assume that admissible surfaces for $\operatorname{scl}$
have certain topological properties:

######  Definition 2.3.

An admissible surface $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ is said to
be

  * •

_Disc- and sphere-free_ if no connected component of $\Sigma$ is a disc or a
sphere,

  * •

_Incompressible_ if every simple closed curve in $\Sigma$ with nullhomotopic
image in $X$ is nullhomotopic in $\Sigma$,

  * •

_Monotone_ if the restriction of $\partial f:\partial\Sigma\rightarrow S^{1}$
to every boundary component of $\Sigma$ has positive degree.

######  Proposition 2.4 (Reduction).

Consider an admissible surface $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$
with $\partial f_{*}\left[\partial\Sigma\right]=n(\Sigma)\left[S^{1}\right]$
for some $n(\Sigma)\in\mathbb{N}_{\geq 1}$.

Then there is a monotone, incompressible, disc- and sphere-free admissible
surface
$f^{\prime}:\left(\Sigma^{\prime},\partial\Sigma^{\prime}\right)\rightarrow(X,g)$
with $\partial
f^{\prime}_{*}\left[\partial\Sigma^{\prime}\right]=n\left(\Sigma^{\prime}\right)\left[S^{1}\right]$
for some $n\left(\Sigma^{\prime}\right)\in\mathbb{N}_{\geq 1}$, and such that

$\frac{-\chi^{-}\left(\Sigma^{\prime}\right)}{2n\left(\Sigma^{\prime}\right)}\leq\frac{-\chi^{-}\left(\Sigma\right)}{2n(\Sigma)}.$

###### Proof.

To obtain a disc- and sphere-free surface, it suffices to throw away any disc
or sphere component as this does not change the reduced Euler characteristic.
For incompressibility, the idea is that one can glue a disc onto a non-
contractible simple closed curve with nullhomotopic image, thereby reducing
its complexity. For monotonicity, see for example [cal-scl]*Proposition 2.13.
∎

######  Proposition 2.5 (Transversality).

Let $X$ be a connected $2$-complex, $g\in\pi_{1}X$. Then any incompressible
admissible surface $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ may be
homotoped to one that is _transverse_ , in the sense that it has a
decomposition into the following pieces:

  * •

_Vertex discs_ , i.e. discs mapping to vertices of $X$,

  * •

_$1$ -handles_, i.e. trivial $I$-bundles over edges of $X$, and

  * •

_Cellular discs_ , i.e. discs mapping homeomorphically onto $2$-cells of $X$.

###### Proof.

This follows from Buoncristiano, Rourke and Sanderson’s Transversality Theorem
[brs]*§VII.2. ∎

Propositions 2.4 and 2.5 say that, for the purpose of computing
$\operatorname{scl}$, we can restrict our attention to admissible surfaces
which are transverse, disc- and sphere-free, incompressible, and monotone.
Figure 1 represents a part of such a surface.

a cellular disca $1$-handlea vertex disc$\Sigma$ $X$ Figure 1. Part of a
transverse incompressible admissible surface.

${{{\xrightarrow{\phantom{aa}f\phantom{aa}}}}}$

##  3\. Letter-quasimorphisms

Letter-quasimorphisms were introduced by Heuer [heuer] as a means of obtaining
spectral gaps for $\operatorname{scl}$ in certain amalgams and right-angled
Artin groups.

We denote by $\mathcal{A}$ the set of _alternating words_ in the free group
$F_{2}=F(a,b)$:

$\mathcal{A}=\left\\{b_{0}a_{1}b_{1}\cdots a_{\ell}b_{\ell}a_{\ell+1}\in
F_{2}\phantom{\left\\{1,a^{\pm 1}\right\\}}\right.\\\
\left.\>\middle|\>{}\textrm{$a_{i}\in\left\\{a^{\pm 1}\right\\}$,
$b_{i}\in\left\\{b^{\pm 1}\right\\}$, and $b_{0}\in\left\\{1,b^{\pm
1}\right\\}$, $a_{\ell+1}\in\left\\{1,a^{\pm 1}\right\\}$}\right\\}.$

######  Definition 3.1 (Heuer [heuer]).

A _letter-quasimorphism_ on a group $G$ is a map
$\Phi:G\rightarrow\mathcal{A}$ subject to the following conditions:

  1. (i)

$\Phi\left(g^{-1}\right)=\Phi\left(g\right)^{-1}$ for every $g\in G$, and

  2. (ii)

For every $g,h\in G$, either

    1. (a)

$\Phi\left(gh\right)=\Phi\left(g\right)\Phi\left(h\right)$ (see Figure 2(a)),
or

    2. (b)

$\Phi(g)=u^{-1}x_{1}v$, $\Phi(h)=v^{-1}x_{2}w$, and
$\Phi\left(gh\right)^{-1}=w^{-1}x_{3}u$, for some alternating words
$u,v,w\in\mathcal{A}$, and letters $x_{1},x_{2},x_{3}$ such that either
$x_{1},x_{2},x_{3},x_{1}x_{2}x_{3}\in\left\\{a^{\pm 1}\right\\}$, or
$x_{1},x_{2},x_{3},x_{1}x_{2}x_{3}\in\left\\{b^{\pm 1}\right\\}$ (see Figure
2(b)).

$\Phi(g)$$\Phi(h)$$\Phi(gh)$$u$$v$$w$ (a) Degenerate case.

$x_{1}$$x_{2}$$x_{3}$$u$$v$$w$$\Phi(g)$$\Phi(h)$$\Phi(gh)$ (b) Non-degenerate
case.

Figure 2. Letter-quasimorphism condition.

######  Remark 3.2.

Let $\Phi:G\rightarrow\mathcal{A}$ be a letter-quasimorphism and let $g,h\in
G$ be such that $\Phi(gh)=\Phi(g)\Phi(h)$. Write $\Phi(g)=u^{-1}v$,
$\Phi(h)=v^{-1}w$, and $\Phi(gh)^{-1}=w^{-1}u$ as reduced words, as in Figure
2(a). Then in fact one of $u$, $v$, and $w$ must be trivial. Indeed,
otherwise, since $\Phi(g)$ is a reduced alternating word, if for example the
first letter of $u$ is $a$ or $a^{-1}$, then the first letter of $v$ is $b$ or
$b^{-1}$. Similarly, $\Phi(h)$ is an alternating word, so the first letter of
$w$ must be $a$ or $a^{-1}$. But now both $u$ and $w$ start with $a$ or
$a^{-1}$, which contradicts $\Phi(gh)$ being reduced and alternating.

We will need groups to have enough letter-quasimorphisms in the following
sense:

######  Definition 3.3.

We will say that a group $G$ is _quasi-residually free_ if for every $g\in G$
with $\operatorname{scl}(g)<\infty$111The condition that
$\operatorname{scl}(g)<\infty$ is equivalent to $g$ vanishing in
$H_{1}\left(G;\mathbb{Q}\right)$, or equivalently, to some power of $g$ lying
in the commutator subgroup., there is a letter-quasimorphism
$\Phi:G\rightarrow\mathcal{A}$ such that $\Phi(g)\neq 1$, and
$\Phi\left(g^{n}\right)=\Phi(g)^{n}$ for all $n\in\mathbb{Z}$.

######  Example 3.4 (Heuer [heuer]).

The following groups are quasi-residually free:

  1. (i)

Residually free groups [heuer]*Example 4.2,

  2. (ii)

Right-angled Artin groups [heuer]*Theorem 7.2.

Heuer [heuer] showed that quasi-residually free groups have a strong spectral
gap for $\operatorname{scl}$. To do this, he explains how one can use a
letter-quasimorphism $G\rightarrow\mathcal{A}$ to construct a classical, real-
valued quasimorphism $G\rightarrow\mathbb{R}$; he then uses Bavard Duality to
obtain lower bounds on $\operatorname{scl}$.

We take a different route and show that a letter-quasimorphism gives rise to
an angle structure on admissible surfaces, which can in turn be used to
estimate the Euler characteristic. This approach is closer in spirit to Duncan
and Howie’s strong spectral gap in free groups [duncan-howie].

##  4\. Angle structures

We will work with combinatorial $2$-dimensional CW-complexes, which we call
_$2$ -complexes_ for short. Combinatoriality means that the boundary of each
cell has a subdivision for which the restriction of the attaching map to each
subcell is a homeomorphism onto its image. This follows Gersten’s terminology
[gersten].

###  4.a. Definition and Gauß–Bonnet Formula

Wise [wise:ang] introduced angle structures to study the interplay of
geometric and topological properties of $2$-complexes. The idea is to assign a
numerical angle to each corner of the $2$-complex of interest.

Let $X$ be a $2$-complex. A _corner_ of a face $f$ at a vertex $v$ is an edge
in the link $\operatorname{Lk}_{X}(v)$ of $v$ corresponding to the face $f$.
We denote by $\mathcal{C}(f)$ the set of corners of the face $f$ and by
$\mathcal{C}(v)$ the set of corners at the vertex $v$.

######  Definition 4.1 (Wise [wise:ang]).

An _angle structure_ on $X$ is the assignment of a real number $\angle c$ to
each corner $c$ in $X$. We then say that $X$ is an _angled $2$-complex_.

Associated to an angle structure, there is a notion of curvature:

  * •

Let $f$ be a face of $X$. The _curvature_ of $f$ is

$\kappa(f)=2\pi+\sum_{c\in\mathcal{C}(f)}\left(\angle c-\pi\right).$

  * •

Let $v$ be a vertex of $X$. The _curvature_ of $v$ is

$\kappa(v)=2\pi-\pi\cdot\chi\left(\operatorname{Lk}_{X}(v)\right)-\sum_{c\in\mathcal{C}(v)}\angle
c.$

An angle structure on $X$ is said to be _non-positively curved_ if
$\kappa(f)\leq 0$ for every face $f$ of $X$ and $\kappa(v)\leq 0$ for every
vertex $v$ of $X$.

The _total curvature_ of an angled $2$-complex $X$ is the quantity

$\kappa(X)=\sum_{f\in F(X)}\kappa(f)+\sum_{v\in V(X)}\kappa(v),$

where $F(X)$ and $V(X)$ are the face set and vertex set of $X$, respectively.

Angle structures arise naturally — a prominent example is that of
$\operatorname{CAT}(0)$ cell complexes [bh]*Chapter II.3. The natural angle
structure on a $\operatorname{CAT}(0)$ cell complex is non-positively curved.

The relevance of angle structures comes from the _Combinatorial Gauß–Bonnet
Formula_ , which constrains the topology of a $2$-complex depending on its
total curvature.

######  Proposition 4.2 (Combinatorial Gauß–Bonnet).

Let $X$ be an angled $2$-complex. Then

$2\pi\cdot\chi(X)=\kappa(X).$

###### Proof.

By definition, the total curvature $\kappa(X)$ is equal to

$\sum_{f\in F(X)}\left(2\pi+\sum_{c\in\mathcal{C}(f)}\left(\angle
c-\pi\right)\right)+\sum_{v\in
V(X)}\left(2\pi-\pi\cdot\chi\left(\operatorname{Lk}_{X}(v)\right)-\sum_{c\in\mathcal{C}(v)}\angle
c\right).$

Note that every corner appears as a corner of a unique face at a unique
vertex, so that

(*) $\sum_{f\in F(X)}\sum_{c\in\mathcal{C}(f)}\angle c=\sum_{v\in
V(X)}\sum_{c\in\mathcal{C}(v)}\angle c.$

Hence,

$\kappa(X)=2\pi\left(\sharp F(X)+\sharp V(X)\right)-\pi\sum_{f\in
F(X)}\sharp\mathcal{C}(f)-\pi\sum_{v\in
V(X)}\chi\left(\operatorname{Lk}_{X}(v)\right).$

Now the link of a vertex $v$ is a graph, with vertices corresponding to half-
edges of $X$ incident to $v$, and edges corresponding to corners of $X$ at
$v$. It follows that

$\sum_{v\in V(X)}\chi\left(\operatorname{Lk}_{X}(v)\right)=2\sharp
E(X)-\sum_{v\in V(X)}\sharp\mathcal{C}(v).$

Similar to $\left(\ref{eq:sums-corners}\right)$, we have
$\sum_{f}\sharp\mathcal{C}(f)=\sum_{v}\sharp\mathcal{C}(v)$. We finally obtain

$\kappa(X)=2\pi\left(\sharp F(X)-\sharp E(X)+\sharp
V(X)\right)=2\pi\cdot\chi(X).\qed$

###  4.b. Interior curvature of surfaces

We have defined angle structures on $2$-complexes, which are constructed by
gluing a collection of discs to a graph. Each disc then has a notion of
curvature as explained above. The objects of which we will want to estimate
the curvature in the sequel will have a more natural decomposition into
compact subsurfaces, rather than discs; this is our motivation for introducing
the following notions.

We consider a compact surface $\Lambda$ with nonempty boundary, with a
cellulation whose vertex set $V(\Lambda)$ is contained in $\partial\Lambda$.
Suppose that $\Lambda$ is equipped with an angle structure.

  * •

The _total angle_ of a vertex $v$ of $\Lambda$ is

$\angle_{\operatorname{tot}}^{\Lambda}(v)=\sum_{c\in\mathcal{C}(v)}\angle c.$

The reason why we include $\Lambda$ in the notation is that we will typically
be considering a decomposition of a $2$-complex into subsurfaces, so that a
vertex can be seen as belonging to one subsurface or another.

  * •

The _interior curvature_ of $\Lambda$ is

$\kappa_{\operatorname{int}}(\Lambda)=2\pi\cdot\chi(\Lambda)+\sum_{v\in
V(\Lambda)}\left(\angle_{\operatorname{tot}}^{\Lambda}(v)-\pi\right).$

Note in particular that if $\Lambda$ is a disc, then $\chi(\Lambda)=1$ and
$\kappa_{\operatorname{int}}(\Lambda)$ is the curvature of $\Lambda$ seen as a
face.

We’ll use the following elementary observation:

######  Lemma 4.3.

Let $\Lambda$ be a compact surface equipped with an angle structure supported
on a cellulation of $\Lambda$ with $V(\Lambda)\subseteq\partial\Lambda$. Then

$\kappa_{\operatorname{int}}(\Lambda)=\sum_{f\in F(\Lambda)}\kappa(f),$

where $F(\Lambda)$ is the face set of $\Lambda$.

###### Proof.

Each vertex $v\in V(\Lambda)$ lies on $\partial\Lambda$, so its link is
homeomorphic to a line segment, and we have

$\kappa(v)=2\pi-\pi\cdot\chi\left(\operatorname{Lk}_{\Lambda}(v)\right)-\sum_{c\in\mathcal{C}(v)}\angle
c=\pi-\angle_{\operatorname{tot}}^{\Lambda}(v).$

Hence, the Gauß–Bonnet Formula (Proposition 4.2) gives

$\sum_{f\in F(\Lambda)}\kappa(f)=2\pi\cdot\chi(\Lambda)+\sum_{v\in
V(\Lambda)}\left(\angle_{\operatorname{tot}}^{\Lambda}(v)-\pi\right)=\kappa_{\operatorname{int}}(\Lambda).\qed$

This says that, if $2$-cells — which are normally discs — are replaced with
more general compact surfaces with nonempty boundary, then the interior
curvature is the right analogue of the curvature of a face.

######  Corollary 4.4.

Let $X$ be an angled $2$-complex. Suppose that there is a finite collection
$\left(\Lambda_{i}\right)_{i\in I}$ of compact surfaces with nonempty boundary
embedded in $X$, such that

  * •

Each face of $X$ is contained in a unique $\Lambda_{i}$, and

  * •

Each vertex of $X$ is contained in $\partial\Lambda_{i}$ for some (possibly
several) $i\in I$.

Then the total curvature of $X$ can be computed via

$\kappa(X)=\sum_{i\in
I}\kappa_{\operatorname{int}}\left(\Lambda_{i}\right)+\sum_{v\in
V(X)}\kappa(v).\qed$

##  5\. From letter-quasimorphisms to angle structures

We now reach the core of this paper. The goal is to prove the following:

######  Theorem A.

Let $X$ be a connected $2$-complex and let $g\in\pi_{1}X\smallsetminus 1$.
Assume that there is a letter-quasimorphism
$\Phi:\pi_{1}X\rightarrow\mathcal{A}$ with $\Phi(g)\neq 1$ and
$\Phi\left(g^{n}\right)=\Phi(g)^{n}$ for all $n\in\mathbb{Z}$.

Then given a monotone, incompressible, disc- and sphere-free admissible
surface $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ with
$f_{*}\left[\partial\Sigma\right]=n(\Sigma)\left[S^{1}\right]$ for some
$n(\Sigma)\in\mathbb{N}_{\geq 1}$, there is an angle structure on $\Sigma$
whose total curvature satisfies

$\kappa\left(\Sigma\right)\leq-2\pi\cdot n(\Sigma).$

Before entering the proof of Theorem A, we explain its implications for
$\operatorname{scl}$:

######  Corollary 5.1 (Heuer [heuer]*Theorem 4.7).

Let $G$ be a group and $g\in G\smallsetminus 1$. Assume that there is a
letter-quasimorphism $\Phi:G\rightarrow\mathcal{A}$ with $\Phi(g)\neq 1$ and
$\Phi\left(g^{n}\right)=\Phi(g)^{n}$ for all $n\in\mathbb{Z}$.

Then $\operatorname{scl}(g)\geq\frac{1}{2}$.

###### Proof.

Fix $X$ a connected $2$-complex with $\pi_{1}X=G$ (for example, $X$ can be a
presentation $2$-complex of $G$). Proposition 2.4 says that
$\operatorname{scl}$ can be computed as the infimum of
$\frac{-\chi^{-}(\Sigma)}{2n(\Sigma)}$ over all disc- and sphere-free,
incompressible, monotone admissible surfaces. For any such admissible surface
$\Sigma$, Theorem A gives an angle structure on $\Sigma$ with a bound on the
total curvature. Now the Gauß–Bonnet Formula (Proposition 4.2) translates the
bound on $\kappa\left(\Sigma\right)$ into a bound on
$\chi\left(\Sigma\right)$, and one obtains

$-\chi^{-}(\Sigma)\geq-\chi(\Sigma)=-\frac{1}{2\pi}\kappa\left(\Sigma\right)\geq
n(\Sigma).$

After passing to the infimum, this implies that
$\operatorname{scl}(g)\geq\frac{1}{2}$. ∎

######  Corollary 5.2 (Heuer [heuer]).

Every quasi-residually free group has a strong spectral gap for
$\operatorname{scl}$.∎

The rest of this section is devoted to proving the main theorem.

###### Proof of Theorem A.

Note that $\Phi(g)$ is an alternating word in $F_{2}=F(a,b)$, and let us
assume to fix notations that it starts with $a$ or $a^{-1}$ and ends with $b$
or $b^{-1}$, so that

$\Phi(g)=a_{1}b_{1}\cdots a_{\ell}b_{\ell},$

with $a_{i}\in\left\\{a^{\pm 1}\right\\}$ and $b_{i}\in\left\\{b^{\pm
1}\right\\}$.

Start with $f:(\Sigma,\partial\Sigma)\rightarrow(X,g)$ a disc- and sphere-
free, incompressible, monotone admissible surface with
$f_{*}\left[\partial\Sigma\right]=n(\Sigma)\left[S^{1}\right]$ for some
$n(\Sigma)\in\mathbb{N}_{\geq 1}$.

After contracting a spanning tree in the $1$-skeleton, one may assume that $X$
has only one vertex. One may also subdivide the faces of $X$ to ensure that
they are all triangles. After applying Proposition 2.5, we can further assume
that $f$ is transverse.

###  5.a. Subdividing $1$-handles via the letter-quasimorphism

Each $1$-handle $\mathcal{H}$ in $\Sigma$ is a trivial $I$-bundle over an edge
$e$ of $X$, which is a loop and represents an element
$g_{\mathcal{H}}\in\pi_{1}X$. We consider the image of $g_{\mathcal{H}}$ under
our letter-quasimorphism $\Phi:\pi_{1}X\rightarrow\mathcal{A}$; this is an
alternating word in $F_{2}$, and we write for example

$\Phi\left(g_{\mathcal{H}}\right)=\alpha_{1}\beta_{1}\cdots\alpha_{k}\beta_{k},$

with $\alpha_{i}\in\left\\{a^{\pm 1}\right\\}$ and $\beta_{i}\in\left\\{b^{\pm
1}\right\\}$. The element $\Phi\left(g_{\mathcal{H}}\right)\in F_{2}$ is
represented by an immersed loop
$\gamma_{\Phi\left(g_{\mathcal{H}}\right)}:S^{1}\looparrowright B_{2}$, where
$B_{2}=S^{1}_{a}\vee S^{1}_{b}$ is a bouquet of two oriented circles labelled
by $a$ and $b$.

The $1$-handle $\mathcal{H}$ being a trivial $I$-bundle over $e$ means that it
is homeomorphic to the product $I\times e$ (where $I=[0,1]$), and the map
$\mathcal{H}\rightarrow e$ is given by projection onto the second coordinate.
Now pick an orientation-preserving homeomorphism $e\cong S^{1}$ and consider
the composition

$\mathcal{H}=I\times e\xrightarrow{\textnormal{proj}_{2}}e\cong
S^{1}\xrightarrow{\gamma_{\Phi\left(g_{\mathcal{H}}\right)}}B_{2}=S^{1}_{a}\vee
S^{1}_{b}.$

This map is transverse, and defines a decomposition of $\mathcal{H}$ into $2k$
$1$-handles mapping successively to
$\alpha_{1},\beta_{1},\dots,\alpha_{k},\beta_{k}$. To distinguish those
$1$-handles from the original ones, we call them _stripes_. Stripes can be
divided into two types:

  * •

_$a$ -stripes_, with image contained in $S^{1}_{a}$, and

  * •

_$b$ -stripes_, with image contained in $S^{1}_{b}$.

We’ll depict those two types of stripes with two different colours (red and
green). The map from each stripe to $S^{1}_{a}$ or $S^{1}_{b}$ will be encoded
in pictures by arrows parallel to the base edge of the $I$-bundle, indicating
the positive direction of $S^{1}_{a}$ or $S^{1}_{b}$ — see Figure 3.

$g_{\mathcal{H}}$$\mathcal{H}$
$b$$b$$b$$a$$a$$a$$\Phi\left(g_{\mathcal{H}}\right)$ Figure 3. Decomposition
of a $1$-handle into stripes via the letter-quasimorphism. Here,
$\Phi\left(g_{\mathcal{H}}\right)=ab^{-1}a^{-1}b^{-1}ab$.

${{{{\rightsquigarrow}}}}$

###  5.b. Extension to cellular discs

The above construction defines a new cellular structure on the $1$-handles of
$\Sigma$, and we now want to do something similar to cellular discs. At the
end of the construction, $\Sigma$ will be decomposed into its preexisting
vertex discs, $a$-regions (extending $a$-stripes), and $b$-regions (extending
$b$-stripes).

Each cellular disc $\mathcal{D}$ maps homeomorphically to a (triangular)
$2$-cell of $X$, and has three sides mapping under $f$ to edges representing
elements $g_{1},g_{2},g_{3}\in\pi_{1}X$. The presence of the $2$-cell means
that $g_{3}=\left(g_{1}g_{2}\right)^{-1}$. Now the three $1$-handles incident
to $\mathcal{D}$ have been subdivided as explained above into parallel stripes
labelled by $a^{\pm 1}$ and $b^{\pm 1}$, and the concatenation of those labels
along each of the three edges of the triangle are given by
$\Phi\left(g_{1}\right)$, $\Phi\left(g_{2}\right)$, and
$\Phi\left(g_{1}g_{2}\right)^{-1}$. By the letter-quasimorphism condition,
they satisfy one of the patterns of Figure 2: we can write

$\Phi\left(g_{1}\right)=u^{-1}x_{1}v,\quad\Phi\left(g_{2}\right)=v^{-1}x_{2}w,\quad\Phi\left(g_{1}g_{2}\right)^{-1}=w^{-1}x_{3}u,$

with either $x_{1}=x_{2}=x_{3}=1$, or
$x_{1},x_{2},x_{3},x_{1}x_{2}x_{3}\in\left\\{a^{\pm 1}\right\\}$, or
$x_{1},x_{2},x_{3},x_{1}x_{2}x_{3}\in\left\\{b^{\pm 1}\right\\}$. Moreover,
Remark 3.2 says that, in the degenerate case (when $x_{1}=x_{2}=x_{3}=1$), we
have $u=1$ or $v=1$ or $w=1$ — we will assume that $w=1$ to fix notations.

As illustrated by Figure 4, the boundary of $\mathcal{D}$ has two consecutive
sections labelled respectively by $u$ and $u^{-1}$, and these sections lie on
different sides of $\mathcal{D}$. We can connect the vertices of these two
sections and extend the $a$\- and $b$-stripes across part of $\mathcal{D}$. We
then apply the same process to $v$ and $w$. The new stripes that we have
constructed inside $\mathcal{D}$ are called $a$\- or $b$-stripes depending on
whether they extend $a$\- or $b$-stripes. Note that some of the new stripes,
at the vertices of the cellular disc, are triangular (one of the fibres of the
$I$-bundle is collapsed to a point), but we still call them stripes.

In the degenerate case (Figure 4(a)), the new stripes now fill $\mathcal{D}$.
In the non-degenerate case (Figure 4(b)), we are left with one unfilled
hexagon inside $\mathcal{D}$; this hexagon has three sides lying on
$\partial\mathcal{D}$ and mapping either all to $S^{1}_{a}$ or all to
$S^{1}_{b}$. We declare the hexagon to be of type $a$ or $b$ accordingly.

$u$$v$$u$$v$ (a) Degenerate case.

$u$$x_{3}$$w$$w$$x_{2}$$v$$u$$x_{1}$$v$ (b) Non-degenerate case.

Figure 4. Extension of stripes across cellular discs.

This construction yields a new cellular structure on $\Sigma$, with a
decomposition into vertex discs (which remain unchanged), $a$\- and
$b$-stripes, and $a$\- and $b$-hexagons — see Figure 5. We call this data a
_stripe pattern_.

It is important to note that the new cellular structure does not correspond to
any transverse map to a bouquet of two circles: such a map can be defined on
$a$\- and $b$-stripes as explained above, but it cannot be extended to
hexagons since their boundary maps to a non-trivial element of $F_{2}$ (either
$a^{\pm 1}$ or $b^{\pm 1}$).

an $a$-hexagonan $a$-stripea $b$-stripea vertex disc Figure 5. Part of a
stripe pattern.

###  5.c. Boundary labelling

We now turn our attention to the boundary of $\Sigma$. The (oriented) boundary
edges of $\Sigma$ are either part of a vertex disc, or they bound a stripe and
are then labelled by some letter in $\left\\{a^{\pm 1},b^{\pm 1}\right\\}$.
Moreover, the concatenation of the oriented labels of the boundary edges along
a $1$-handle $\mathcal{H}$ of $\Sigma$ mapping via $f$ to $g_{\mathcal{H}}$ is
$\Phi\left(g_{\mathcal{H}}\right)$ (see Figure 3). Now pick a boundary
component $\partial_{j}$ of $\Sigma$. The image of $\partial_{j}$ under $f$
represents a word $g_{1}\cdots g_{\ell}=g^{k}$ for some $k\in\mathbb{N}_{\geq
1}$, where each $g_{i}$ is an element in $\pi_{1}X$ represented by a single
edge loop in $X$, and $g$ is the element of $\pi_{1}X$ with respect to which
$f:\Sigma\rightarrow X$ is an admissible surface. It follows that the
labelling of $\partial_{j}$ obtained by concatenating the labels of the $a$\-
and $b$-stripes along it is
$\Phi\left(g_{1}\right)\cdots\Phi\left(g_{\ell}\right)$.

We want to modify $\Sigma$ so that the labelling of $\partial_{j}$ becomes
$\Phi\left(g_{1}\cdots g_{\ell}\right)=\Phi\left(g\right)^{k}$, and we will
use the letter-quasimorphism condition to do so. Consider two successive
sections of $\partial_{j}$ that are labelled by $\Phi\left(g_{i}\right)$ and
$\Phi\left(g_{i+1}\right)$. Definition 3.1 says that either
$\Phi\left(g_{i}\right)\Phi\left(g_{i+1}\right)=\Phi\left(g_{i}g_{i+1}\right)$
(in the degenerate case), or there are alternating words $u,v,w\in F_{2}$, and
letters $x_{1},x_{2},x_{3}$ in $\left\\{a^{\pm 1}\right\\}$ or $\left\\{b^{\pm
1}\right\\}$ such that

$\Phi\left(g_{i}\right)=u^{-1}x_{1}v,\quad\Phi\left(g_{i+1}\right)=v^{-1}x_{2}w,\quad\Phi\left(g_{i}g_{i+1}\right)^{-1}=w^{-1}x_{3}u.$

We can use this data to glue some new stripes and (in the non-degenerate case)
one $a$\- or $b$-hexagon to $\partial_{j}$ as in Figure 6, modifying $\Sigma$
by a homeomorphism so that the portion of $\partial_{j}$ that was labelled by
$\Phi\left(g_{i}\right)\Phi\left(g_{i+1}\right)$ is now labelled by
$\Phi\left(g_{i}g_{i+1}\right)$.

$\Phi\left(g_{i}\right)$$\Phi\left(g_{i+1}\right)$$\Phi\left(g_{i}g_{i+1}\right)$
Figure 6. Correcting the boundary labelling. (The former boundary is depicted
as a dotted line and the new one as a full line.)

By repeating this process a finite number of times (as most as many times as
the number of $1$-handles of $\Sigma$), we obtain a surface homeomorphic to
$\Sigma$ with a new stripe pattern, and whose boundary components are all
labelled by some positive power of $\Phi(g)$.

Since boundary components are labelled by powers of $\Phi(g)=a_{1}b_{1}\cdots
a_{\ell}b_{\ell}$, with $a_{i}\in\left\\{a^{\pm 1}\right\\}$ and
$b_{i}\in\left\\{b^{\pm 1}\right\\}$, each boundary edge bounding an
$a$-stripe corresponds to some $a_{i}$ and each boundary edge bounding a
$b$-stripe corresponds to some $b_{i}$. We will remember the index $i$ as part
of the stripe pattern. Hence, reading the successive indices of the edges
along a boundary component of $\Sigma$ yields a cyclic permutation of a
positive iterate of the sequence $\left(1,1,\dots,\ell,\ell\right)$.

######  Example 5.3.

Suppose that $X$ is a bouquet of two oriented circles labelled by $a$ and $b$,
so that $\pi_{1}X=F(a,b)$. Let $g=[a,b]=aba^{-1}b^{-1}\in\pi_{1}X$. There is a
letter-quasimorphism $\Phi:\pi_{1}X\rightarrow\mathcal{A}$ given by

$a^{m_{1}}b^{n_{1}}\cdots a^{m_{k}}b^{n_{k}}\longmapsto
a^{\operatorname{sign}\left(m_{1}\right)}b^{\operatorname{sign}\left(n_{1}\right)}\cdots
a^{\operatorname{sign}\left(m_{k}\right)}b^{\operatorname{sign}\left(n_{k}\right)},$

with $m_{i},n_{i}\in\mathbb{Z}$, all non-zero except possibly for $m_{1}$ and
$n_{k}$ (see [heuer]*Example 4.2).

Applying the above construction to the admissible surface
$f:\left(\Sigma,\partial\Sigma\right)\rightarrow\left(X,g\right)$ which is a
once-punctured torus with generators mapping to $X$ in the standard way yields
the stripe pattern of Figure 7. The only boundary component of $\Sigma$ is
labelled by $\left(1,1,2,2\right)$.

$1$$2$$2$$1$$\circlearrowleft$ Figure 7. Stripe pattern on an admissible
surface for $g=[a,b]\in F(a,b)$ (black arrows indicate edge identifications).

###  5.d. Unzipping $\Sigma$

With its current stripe pattern, $\Sigma$ is subdivided into:

  * •

The union $\Sigma^{D}$ of its closed vertex discs,

  * •

The union $\Sigma^{a}$ of its closed $a$-stripes and $a$-hexagons, and

  * •

The union $\Sigma^{b}$ of its closed $b$-stripes and $b$-hexagons.

Consider the boundary of those different regions:

$\Gamma=\left(\Sigma^{a}\cap\Sigma^{b}\right)\cup\left(\Sigma^{a}\cap\Sigma^{D}\right)\cup\left(\Sigma^{b}\cap\Sigma^{D}\right).$

By observing the local structure of the stripe pattern, one can see that
$\Gamma$ is an embedded graph in $\Sigma$.

We want to perform one last modification on the stripe pattern to remove all
singular points of $\Gamma$ (i.e. to ensure that $\Gamma$ is a $1$-dimensional
submanifold of $\Sigma$). Note that the only points where $\Gamma$ might not
be locally homeomorphic to a line are the apices of degenerate cellular discs
(see Figure 4(a)). We can _unzip_ each degenerate cellular disc as indicated
in Figure 8, extending an existing vertex disc between the $a$\- and
$b$-stripes meeting at the apex of the triangle.

Figure 8. Unzipping a degenerate cellular disc.

${{{{\rightsquigarrow}}}}$

After unzipping one cellular disc, one needs to keep unzipping along the same
leaf of $\Gamma$. One can iterate until there is no more degenerate cellular
disc. After this process, $\Gamma$ is locally homeomorphic to a line.
Moreover, $\Gamma$ is compact as its intersection with each vertex disc,
$1$-handle, or cellular disc of $\Sigma$ is compact, and $\Sigma$ has only
finitely many such pieces. Therefore, $\Gamma$ is a compact $1$-dimensional
submanifold of $\Sigma$. We then say that the stripe pattern is an _unzipped
stripe pattern_.

###  5.e. The angle structure

Before constructing the angle structure on $\Sigma$, we describe the
cellulation that will support it.

An _$a$ -region_ of $\Sigma$ is a connected component of $\Sigma^{a}$, while a
_$b$ -region_ is a connected component of $\Sigma^{b}$. A _region_ is either
an $a$-region, a $b$-region, or a vertex disc. A _transition arc_ is a
connected component of the $1$-dimensional submanifold $\Gamma$ discussed
above, while a _boundary arc_ is an edge of $\partial\Sigma$ that bounds a
stripe. Recall that each boundary arc has an index
$i\in\left\\{1,\dots,\ell\right\\}$ such that the arc is labelled by the
letter $a_{i}\in\left\\{a^{\pm 1}\right\\}$ (or $b_{i}\in\left\\{b^{\pm
1}\right\\}$) of the word $\Phi(g)$.

Observe that each $a$\- or $b$-region is a subsurface of $\Sigma$, and each of
its boundary components alternates between transition and boundary arcs. We
put one vertex at each endpoint of each transition or boundary arc — those
vertices are called _arc endpoints_ — and an additional vertex in the interior
of each transition arc — those are called _transition vertices_. Then we add
boundary arcs and half-transition arcs as edges. In order to complete this to
a cellulation of $\Sigma$, we need to add more edges (until the edge set cuts
$\Sigma$ into discs), but in light of Corollary 4.4, the way we do this is
irrelevant as we will estimate the total curvature of $\Sigma$ by counting the
interior curvature of each region, and the interior curvature does not depend
on how a subsurface is subdivided into discs. Likewise, it suffices to define
the total angle of each vertex inside each region, without specifying how the
total angle is split between the faces of $\Sigma$.

The angle structure is now defined as follows.

Each arc endpoint $v$ is contained in two regions (on both sides of the
corresponding transition arc), or exceptionally one if $v$ is the endpoint of
a boundary arc which is a loop. The total angle of $v$ in each of these
regions is defined to be

  * •

A right angle ($\pi/2$) if $v$ is contained in two regions and
$v\in\partial\Sigma$, or

  * •

A flat angle ($\pi$) if $v$ is contained in two regions and
$v\not\in\partial\Sigma$, or if $v$ is contained in only one region.

Consider a transition vertex $v$. It lies in two regions, at most one of which
is a vertex disc. Consider an $a$\- or $b$-region $\Lambda$ on the boundary of
which $v$ lies. Recall that the boundary of $\Lambda$ alternates between
transition and boundary arcs; $v$ lies on a transition arc, which is preceded
by a boundary arc with index $i\in\left\\{1,\dots,\ell\right\\}$, and
succeeded by another boundary arc with index
$j\in\left\\{1,\dots,\ell\right\\}$ — here, the order is induced by the
orientation that $\Lambda$ inherits from $\Sigma$. We define the total angle
of $v$ in the $a$\- or $b$-region $\Lambda$ to be

(†)
$\angle_{\operatorname{tot}}^{\Lambda}(v)=\theta_{i,j}=\begin{cases}2\pi\quad&\textrm{if
$i\geq j$}\\\ 0\quad&\textrm{if $i<j$}\end{cases}.$

In the exceptional case where $v$ is the endpoint of a transition arc which is
a loop, there is no preceding or succeeding boundary arc, and we set
$\angle_{\operatorname{tot}}^{\Lambda}(v)=\pi$.

Figure 9 shows an annular $a$-region and the definition of the angle structure
in that region.

If the other region in which $v$ lies is also an $a$\- or $b$-region, then its
total angle in that region is defined in the same way. Otherwise, $v$ lies in
a vertex disc $\Delta$, and its total angle in $\Delta$ is defined to be

(‡)
$\angle_{\operatorname{tot}}^{\Delta}(v)=2\pi-\angle_{\operatorname{tot}}^{\Lambda}(v).$

${i_{1}}$${i_{2}}$${i_{3}}$${i_{4}}$${i_{5}}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\theta_{i_{3},i_{1}}}$$\bm{\theta_{i_{1},i_{2}}}$$\bm{\theta_{i_{2},i_{3}}}$$\bm{\theta_{i_{4},i_{5}}}$$\bm{\theta_{i_{5},i_{4}}}$${\circlearrowleft}$an
arc endpointa transition vertexa transition arca boundary arc Figure 9. The
angle structure in an $a$-region. Boundary arcs are in bold, with labels
indicated by parallel arrows. Angles are indicated in purple.

###  5.f. Boundary orientation of $a$\- and $b$-regions

In order to estimate the interior curvature of $a$\- and $b$-regions, we need
a better understanding of their boundary labelling. Each boundary arc of
$\Sigma$ is labelled by a letter in $\left\\{a^{\pm 1},b^{\pm 1}\right\\}$.
Another way to see this labelling is as an orientation of the boundary arc, as
indicated by bold arrows in Figures 4, 5, 6, 7, 8, 9 — the label of the
boundary arc is $a^{+1}$ (or $b^{+1}$) if its orientation matches that of
$\Sigma$, and $a^{-1}$ (or $b^{-1}$) otherwise. We say that two boundary arcs
have _opposite orientations_ if one of them is labelled by $a^{+1}$ (or
$b^{+1}$) while the other one is labelled by $a^{-1}$ (or $b^{-1}$) — in other
words, one is oriented consistently with $\Sigma$, and the other one isn’t.

######  Lemma 5.4.

Let $\Lambda$ be an $a$\- or $b$-region of $\Sigma$ whose boundary contains at
least two distinct boundary arcs. Then $\partial\Lambda$ contains two boundary
arcs with opposite orientations.

###### Proof.

The region $\Lambda$ is built out of stripes and hexagons. Each stripe has
four boundary edges, alternating between sections of transition arcs, and
edges that have an orientation given by whether they map positively to
$S^{1}_{a}$ or $S^{1}_{b}$. Those edges are called _directed edges_. The
orientations of directed edges are indicated by bold arrows in Figures 4, 5,
6, 7, 8, 9. Likewise, each hexagon has six boundary edges, alternating between
sections of transitions arcs, and directed edges. Stripes and hexagons are
glued along directed edges, in such a way that orientations of directed edges
which are glued together agree.

Now the key observation is that each stripe and each hexagon contains two
directed edges with opposite orientations (i.e. one of them matches the
orientation of the stripe or hexagon, while the other does not). In stripes
(Figure 10(a)), this is because parallel directed edges have parallel
orientations, so one of them matches the orientation of the stripe, and the
other does not. In hexagons (Figure 10(b)), one must come back to the
construction of §5.b to see that the labels of the boundary edges correspond
to letters $x_{1},x_{2},x_{3}$ as in the definition of letter-quasimorphisms
(Definition 3.1), and so in particular $x_{1}x_{2}x_{3}\in\left\\{a^{\pm
1},b^{\pm 1}\right\\}$. This means that each hexagon has two directed edges
with opposite orientations.

$\circlearrowleft$ (a) Stripe.

$\circlearrowleft$ (b) Hexagon.

Figure 10. Directed edges with opposite orientations (highlighted in bold) in
stripes and hexagons.

Using this observation, and the fact that stripes and hexagons are glued so
that directed edges have matching orientations, we can construct two boundary
arcs of $\Lambda$ with opposite orientations as follows. Start at a boundary
arc of $\Lambda$; this arc is a directed edge $e_{1}$ of a stripe or a
hexagon, which we denote by $\Delta_{1}$. We have observed that
$\partial\Delta_{1}$ has a directed edge $e_{2}$ with orientation opposite to
$e_{1}$; pick a point $p_{1}$ in the interior of $e_{1}$ and a point $p_{2}$
in the interior of $e_{2}$ and connect them via an arc in the interior of
$\Delta_{1}$. Now $e_{2}$ borders another stripe or hexagon, which we denote
by $\Delta_{2}$; iterate the above process and extend the previous arc to a
point $p_{3}$ in the interior of a directed edge $e_{3}$ of $\Delta_{3}$ with
orientation opposite to $e_{2}$. Since the region $\Lambda$ is compact, we
must eventually reach a boundary arc of $\Lambda$. The construction is
illustrated in Figure 11.

$\partial\Sigma$$\partial\Sigma$$p_{1}$$p_{2}$$p_{3}$$\eta$$\Lambda$${\circlearrowleft}$
Figure 11. Construction of a path connecting two boundary arcs with opposite
orientations in an $a$\- or $b$-region.

Hence, we obtain an arc $\eta$ with both endpoints on boundary arcs of
$\Lambda$, with the property that all directed edges of stripes or hexagons
crossed by $\eta$ have the same transverse orientation with respect to $\eta$.
In particular, the boundary arcs at the extremities of $\eta$ must have
opposite orientations (with respect to the orientation of $\Lambda$). ∎

###  5.g. Estimating the curvature

We will estimate the total curvature of $\Sigma$ using its decomposition into
$a$-regions, $b$-regions, and vertex discs. Corollary 4.4 says that it
suffices to estimate the curvature of each vertex and the interior curvature
of each region.

######  Claim 1.

Each arc endpoint of $\Sigma$ has zero curvature.

###### Proof of the claim.

Let $v$ be an arc endpoint.

  * •

Suppose that $v$ lies at the endpoint of a transition arc and a boundary arc.
In particular, $v\in\partial\Sigma$, and $v$ is contained in two regions. The
angle of $v$ in each region is $\pi/2$, and
$\chi\left(\operatorname{Lk}_{\Sigma}(v)\right)=1$, so that $\kappa(v)=0$.

  * •

If $v$ is only the endpoint of a boundary arc (which has to be a loop), then
it is contained in a single region, where its angle is $\pi$, so that
$\kappa(v)=0$.

  * •

If $v$ is only the endpoint of a transition arc (which has to be a loop
again), then $v\not\in\partial\Sigma$, $v$ is contained in two regions, and
its angle is $\pi$ in each. Since
$\chi\left(\operatorname{Lk}_{\Sigma}(v)\right)=0$ in this case, we get
$\kappa(v)=0$ again.∎

######  Claim 2.

Each $a$\- or $b$-region of $\Sigma$ has non-positive interior curvature.

###### Proof of the claim.

Let $\Lambda$ be an $a$\- or $b$-region of $\Sigma$. Recall from §4 that the
interior curvature of $\Lambda$ is

$\kappa_{\operatorname{int}}(\Lambda)=2\pi\cdot\chi(\Lambda)+\sum_{v\in
V(\Lambda)}\left(\angle_{\operatorname{tot}}^{\Lambda}(v)-\pi\right).$

We first show that the sum
$S=\sum_{v}\left(\angle_{\operatorname{tot}}^{\Lambda}(v)-\pi\right)$ is non-
positive. To do so, observe that each vertex of $\Lambda$ lies on a unique
transition arc — except in the exceptional case of a vertex $v$ at the
endpoint of a boundary arc which is a loop, but in this case we have
$\left(\angle_{\operatorname{tot}}^{\Lambda}(v)-\pi\right)=0$. Therefore, for
the purpose of computing the sum $S$, we can partition $V(\Lambda)$ into
subsets corresponding to transition arcs. A transition arc which is a loop
contains two vertices of $\Lambda$, each of which has total angle $\pi$ in
$\Lambda$, so that its contribution to $S$ is zero. Each transition arc which
is not a loop contains three vertices of $\Lambda$: two arc endpoints with
total angle $\pi/2$ in $\Lambda$ each, and one transition vertex with total
angle $0$ or $2\pi$ in $\Lambda$ — see Figure 9. Hence, the total contribution
of a transition arc to $S$ is at most
$\left(\frac{\pi}{2}-\pi\right)+\left(\frac{\pi}{2}-\pi\right)+\left(2\pi-\pi\right)=0$.
It follows that $S\leq 0$. This proves that
$\kappa_{\operatorname{int}}(\Lambda)\leq 0$ as soon as $\chi(\Lambda)\leq 0$.

It remains to analyse the case where $\chi(\Lambda)>0$. Since $\Lambda$ has
nonempty boundary, the assumption that $\chi(\Lambda)>0$ implies that
$\Lambda$ is a disc, and $\chi(\Lambda)=1$. Looking back at the previous
argument, the total contribution to the sum $S$ of a transition arc preceded
by a boundary arc with index $i\in\left\\{1,\dots,\ell\right\\}$ and succeeded
by a boundary arc with index $j\in\left\\{1,\dots,\ell\right\\}$ is exactly
$\left(\theta_{i,j}-2\pi\right)$. Now let $i_{1},\dots,i_{k}$ be the
successive indices of the boundary arcs of $\Lambda$, ordered according to the
orientation of $\Lambda$. Then we have

$\kappa_{\operatorname{int}}(\Lambda)=2\pi+\sum_{j=1}^{k}\left(\theta_{i_{j},i_{j+1}}-2\pi\right),$

with the convention that $i_{k+1}=i_{1}$. If the indices $i_{1},\dots,i_{k}$
are not all equal, then there is some $j_{0}$ such that
$i_{j_{0}}<i_{j_{0}+1}$, and so
$\left(\theta_{i_{j_{0}},i_{j_{0}+1}}-2\pi\right)=-2\pi$. Since
$\theta_{i_{j},i_{j+1}}\leq 2\pi$ for all $j$ by definition, it then follows
that

$\kappa_{\operatorname{int}}(\Lambda)\leq
2\pi+\left(\theta_{i_{j_{0}},i_{j_{0}+1}}-2\pi\right)=0.$

Therefore, the only case where $\kappa_{\operatorname{int}}(\Lambda)$ can be
positive is if $\Lambda$ is topologically a disc and all its boundary arcs
have the same index $i$. In particular, all the boundary arcs are labelled by
the same letter (either $a$, $a^{-1}$, $b$, or $b^{-1}$), which contradicts
Lemma 5.4. ∎

######  Claim 3.

The combined contribution of vertex discs and transition vertices to the total
curvature of $\Sigma$ is at most $-2\pi\cdot n(\Sigma)$.

###### Proof of the claim.

Let us start with transition vertices. Each transition vertex $v$ lies on a
transition arc $\alpha$, bounding two distinct regions of $\Sigma$. If
$\alpha$ bounds a vertex disc $\Delta$ and an $a$\- or $b$-region $\Lambda$,
then by $\left(\ref{eq:def-angle-2}\right)$, we have

$\kappa(v)=2\pi-\angle_{\operatorname{tot}}^{\Delta}(v)-\angle_{\operatorname{tot}}^{\Lambda}(v)=0.$

Otherwise, $\alpha$ bounds an $a$-region $\Lambda_{a}$ and a $b$-region
$\Lambda_{b}$. Let $i$ and $j$ be the indices of the boundary arcs preceding
and succeeding $\alpha$ in $\partial\Lambda_{a}$. Hence, the indices of the
boundary arcs preceding and succeeding $\alpha$ in $\partial\Lambda_{b}$ must
be $j-1$ and $i$ (with indices taken modulo $\ell$), as in Figure 12.

$\partial\Sigma$$\partial\Sigma$$i$$j$$i$$j-1$$v$$\alpha$$\bm{\theta_{i,j}}$$\bm{\theta_{j-1,i}}$$\Lambda_{a}$$\Lambda_{b}$$\circlearrowleft$
Figure 12. Estimating the curvature of transition vertices.

It follows that

$\kappa(v)=2\pi-\angle_{\operatorname{tot}}^{\Lambda_{a}}(v)-\angle_{\operatorname{tot}}^{\Lambda_{b}}(v)=2\pi-\theta_{i,j}-\theta_{j-1,i}.$

If $j\neq 1$, then we have $i\geq j$ if and only if $j-1<i$, so that
$\theta_{i,j}=2\pi$ if and only if $\theta_{j-1,i}=0$ (see $\left(\ref{eq:def-
angle}\right)$), and therefore $\kappa(v)=0$.

On the other hand, if $j=1$, then $j-1=\ell$, and we have
$\theta_{i,j}=\theta_{j-1,i}=2\pi$ and $\kappa(v)=-2\pi$.

We now turn our attention to vertex discs. By definition, each vertex disc
$\Delta$ is a topological disc, with boundary alternating between transition
arcs and edges of $\partial\Sigma$ (which are not boundary arcs since they do
not bound a stripe). The successive transition arcs of $\partial\Delta$
alternatively bound an $a$-region $\Lambda_{a}^{(1)}$, then a $b$-region
$\Lambda_{b}^{(1)}$, then another $a$-region $\Lambda_{a}^{(2)}$, etc. Here,
we are ordering the transition arcs in clockwise order — i.e. in the order
opposite to the orientation of $\partial\Delta$. Hence there is an even number
of transition arcs, say $2r$, so that the regions adjacent to $\Delta$ are (in
clockwise order)
$\Lambda_{a}^{(1)},\Lambda_{b}^{(1)},\dots,\Lambda_{a}^{(r)},\Lambda_{b}^{(r)}$.
See Figure 13, where $r=2$.

$\Lambda_{b}^{(1)}$$\Lambda_{a}^{(1)}$$\Lambda_{b}^{(2)}$$\Lambda_{a}^{(2)}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$\bm{\pi/2}$$i_{2}$$i_{2}$$j_{1}-1$$j_{1}$$i_{1}$$i_{1}$$j_{2}-1$$j_{2}$$\bm{2\pi-\theta_{i_{2},j_{2}}}$$\bm{2\pi-\theta_{j_{1}-1,i_{2}}}$$\bm{2\pi-\theta_{i_{1},j_{1}}}$$\bm{2\pi-\theta_{j_{2}-1,i_{1}}}$$\Delta$$\circlearrowleft$$\partial\Sigma$$\partial\Sigma$$\partial\Sigma$$\partial\Sigma$
Figure 13. An octagonal vertex disc.

In $\Lambda_{a}^{(k)}$, let $i_{k}$ and $j_{k}$ be the indices of the boundary
arcs preceding and succeeding the transition arc bounding $\Delta$. Then in
$\Lambda_{b}^{(k)}$, the indices of the boundary arcs preceding and succeeding
the transition arc bounding $\Delta$ must be $j_{k}-1$ and $i_{k+1}$ (with the
convention $i_{r+1}=i_{1}$). Again, we refer the reader to Figure 13 for an
illustration of these notations.

Now $\Delta$ has one transition vertex and two arc endpoints on each
transition arc, and arc endpoints have a total angle of $\pi/2$ in $\Delta$.
Therefore, the interior curvature of $\Delta$ is

(§)
$\kappa_{\operatorname{int}}(\Delta)=2\pi-\sum_{k=1}^{r}\left(\theta_{i_{k},j_{k}}+\theta_{j_{k}-1,i_{k+1}}\right).$

The assumption that $\Sigma$ is disc-free implies that $r\geq 1$. From
$\left(\ref{eq:kappa-int-cell}\right)$, observe first that
$\kappa_{\operatorname{int}}(\Delta)\leq 0$. Indeed, if
$\kappa_{\operatorname{int}}(\Delta)$ were positive, then each term in the sum
would have to equal zero, so that we would have

$i_{k}<j_{k}\quad\textrm{and}\quad j_{k}-1<i_{k+1}$

for all $k$, implying that $i_{1}<j_{1}\leq i_{2}<j_{2}\leq\cdots\leq
i_{r}<j_{r}\leq i_{1}$, which is impossible.

Further, note that each $k\in\left\\{1,\dots,r\right\\}$ such that $j_{k}=1$
adds an extra contribution of $-2\pi$ to the right-hand side of
$\left(\ref{eq:kappa-int-cell}\right)$.

Wrapping everything up, the above computations show that transition vertices
and vertex discs have non-positive (interior) curvature; moreover, every
$a$-boundary arc $\alpha$ with index $1$ will contribute $-2\pi$ to the
curvature

  * •

Of a transition vertex if $\alpha$ is preceded by a $b$-boundary arc along
$\partial\Sigma$, as in Figure 12 with $j=1$, or

  * •

Of a vertex disc if $\alpha$ is preceded by an edge that is part of a vertex
disc, as in Figure 13 with for instance $j_{1}=1$.

Since the indices along each component of $\partial\Sigma$ read
$1,1,\dots,\ell,\ell$, alternating between $a$\- and $b$-boundary arcs, with a
total of $n(\Sigma)$ repetitions of this pattern across all boundary
components of $\Sigma$, there are in total $n(\Sigma)$ $a$-boundary arcs with
index $1$, and their total contribution to the curvature of transition
vertices and vertex discs is at most $-2\pi\cdot n(\Sigma)$ as wanted. ∎

Using the decomposition of $\Sigma$ into vertex discs and $a$\- and
$b$-regions to compute its total curvature via Corollary 4.4 now yields

$\displaystyle\kappa(\Sigma)$
$\displaystyle=\sum_{\begin{subarray}{c}\textrm{$v$ arc}\\\
\textrm{endpoint}\end{subarray}}\overbrace{\kappa(v)}^{=0}+\overbrace{\sum_{\begin{subarray}{c}\textrm{$v$
transition}\\\
\textrm{vertex}\end{subarray}}\kappa(v)+\sum_{\begin{subarray}{c}\textrm{$\Delta$
vertex}\\\
\textrm{disc}\end{subarray}}\kappa_{\operatorname{int}}(\Delta)}^{\leq-2\pi\cdot
n(\Sigma)}+\sum_{\begin{subarray}{c}\textrm{$\Lambda$ $a$- or}\\\
\textrm{$b$-region}\end{subarray}}\overbrace{\kappa_{\operatorname{int}}(\Lambda)}^{\leq
0}$ $\displaystyle\leq-2\pi\cdot n(\Sigma).\qed$

## References