Non-commutative Barge-Ghys quasimorphisms

Michael Brandenbursky,111M. B. acknowledges support of Humboldt foundation.
Misha Verbitsky,222M. V. acknowledges support of HSE University basic research
program, FAPERJ E-26/202.912/2018 and CNPq - Process 310952/2021-2.
2020 Mathematics Subject Classification: 53C05, 20F67, 22E25

Abstract. A (non-commutative) Ulam quasimorphism is a map $q$ from a group
$\Gamma$ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$
belongs to a fixed compact subset of $G$. Generalizing the construction of
Barge and Ghys, we build a family of quasimorphisms on a fundamental group of
a compact manifold $M$ of negative sectional curvature, taking values in an
arbitrary Lie group. This construction, which generalizes the Barge-Ghys
quasimorphisms, associates a quasimorphism to any principal $G$-bundle with
connection on $M$.

Kapovich and Fujiwara have shown that all quasimorphisms taking values in a
discrete group can be constructed from group homomorphisms and quasimorphisms
taking values in a commutative group. We construct Barge-Ghys type
quasimorphisms taking prescribed values on a given subset in $\Gamma$,
producing counterexamples to the Kapovich and Fujiwara theorem for
quasimorphisms taking values in a Lie group. Our construction also generalizes
a result proven by D. Kazhdan in his paper “On $\varepsilon$-representations”.
Kazhdan has proved that for any $\varepsilon>0$, there exists an
$\varepsilon$-representation of the fundamental group of a Riemann surface of
genus 2 which cannot be $1/10$-approximated by a representation. We prove that
this is true for any compact manifold of negative sectional curvature.

###### Contents

  1. 1 Introduction
    1. 1.1 Ulam quasimorphisms
    2. 1.2 Ulam quasimorphisms and stability of representations
    3. 1.3 Quasimorphisms, bounded cohomology and the commutator length
    4. 1.4 Barge-Ghys construction and manifolds of negative curvature
    5. 1.5 Ulam stability and Kazhdan’s $\varepsilon$-representations
    6. 1.6 Constructible quasimorphisms
  2. 2 Barge-Ghys quasimorphisms on fundamental groups of compact manifolds of negative curvature.
    1. 2.1 Manifolds with negative sectional curvature
    2. 2.2 Non-commutative Barge-Ghys quasimorphisms
    3. 2.3 Commutative and non-commutative Barge-Ghys quasimorphisms
  3. 3 Constructible Ulam quasimorphisms
    1. 3.1 Zariski closure of discrete subgroups
    2. 3.2 Infinitely many cyclic subgroups in Gromov hyperbolic groups
    3. 3.3 Non-constructible Ulam quasimorphisms
    4. 3.4 Different notions of quasimorphisms: geometric, algebraic and Ulam
  4. 4 HBG-quasimorphisms
    1. 4.1 Translation length in Gromov hyperbolic groups
    2. 4.2 HBG-quasimorphism associated with a connection
  5. 5 HBG-quasimorphisms with prescribed values
    1. 5.1 Connections with prescribed holonomy
    2. 5.2 Constructing the HBG-quasimorphisms
    3. 5.3 $\varepsilon$-representations

##  1 Introduction

###  1.1 Ulam quasimorphisms

Definition 1.1:  Let $G$ be a Lie group, and $\Gamma$ any group. Ulam
quasimorphism $q:\;\Gamma{\>\longrightarrow\>}G$ is a map which satisfies
$q(x^{-1})=q(x)^{-1}$ and $q(x)q(y)\in K\cdot q(xy)$, where $K\subset G$ is a
fixed compact, independent from the choice of $x,y$. Two Ulam quasimorphisms
$q_{1},q_{2}:\;\Gamma{\>\longrightarrow\>}G$ are called equivalent if there
exists a compact subset $K\subset G$ such that $q_{1}(x)q_{2}(x)^{-1}\in K$,
for all $x\in\Gamma$.

In this paper, we give a versatile construction of Ulam quasimorphisms
associated with connections in vector bundles over a compact manifold $M$ of
negative sectional curvature. As a result, we obtain a quasimorphism from
$\pi_{1}(M)$ to a Lie group, called a Barge-Ghys quasimorphism. This allows us
to find quasimorphisms which are not constructible, in the sense of Fujiwara
and Kapovich [FK]. The Fujiwara-Kapovich constructible quasimorphisms are
obtained from quasimorphisms taking values in abelian groups. Unlike the
constructible quasimorphisms, the Barge-Ghys quasimorphisms we construct
cannot be obtained from ones taking values in abelian groups (3.3).

Interestingly enough, Fujiwara-Kapovich have proved that any Ulam
quasimorphism taking values in a discrete group is constructible. It turns out
that the Ulam quasimorphisms taking values in Lie groups are of entirely
different nature.

As an application, we give a construction of $\varepsilon$-representations
which cannot be approximated by a representation (5.3), generalizing a result
of Kazhdan ([Kaz]).

Note that our definition of Ulam quasimorphism may not be the optimal for some
purposes. Fujiwara and Kapovich [FK] give several non-equivalent (more
relaxed) definitions of a quasimorphism taking values in a non-abelian group
$G$; for abelian $G$, all these definitions are equivalent. Another, even more
relaxed, definition was considered in [HS]. We give a more detailed
presentation of these notions in Subsection 3.4.

###  1.2 Ulam quasimorphisms and stability of representations

The earliest mention of quasimorphisms is found in Ulam’s 1960 book “A
collection of mathematical problems”, Chapter 6 ([U]). Ulam defined an
$\varepsilon$-automorphism of a topological group as a map
$\rho:\;G{\>\longrightarrow\>}G$ such that $\rho(xy)\rho(x)^{-1}\rho(y)^{-1}$
belongs to an $\varepsilon$-neighbourhood of the identity. Ulam asked whether
any such map admits an approximation by an automorphism, in the following
sense: whether for all such $\rho$ there exists a number
$k\in{\mathbb{Z}}^{>0}$ independent from $\rho$ and a homomorphism
$a:\;G{\>\longrightarrow\>}G$ such that for all $x\in G$, the product
$a(x)\rho(x)^{-1}$ belongs to $U$ multiplied by itself $k$ times.

A similar question was considered in 1982 by D. Kazhdan ([Kaz]). He defined an
$\varepsilon$-representation of a group $G$ as a map
$\rho:\;G{\>\longrightarrow\>}U(V)$, where $V$ is a Hilbert space, finite
dimensional or infinite dimensional, satisfying

$\|\rho(xy)\rho(x)^{-1}\rho(y)^{-1}\|<\varepsilon,$

where $\|\cdot\|$ is an operator norm. The distance between two maps
$\rho_{1},\rho_{2}:\;G{\>\longrightarrow\>}U(V)$ can be defined as

$d(\rho_{1},\rho_{2}):=\sup_{x\in G}\|\rho_{1}(x)\rho_{2}(x)^{-1}\|.$

Following a suggestion of V. Milman, Kazhdan asked whether for any $\delta>0$
there exists $\varepsilon>0$ such that any $\varepsilon$-representation can be
$\delta$-approximated by a representation. He proved that this holds true for
amenable groups. When $G=\pi_{1}(S)$, where $S$ is a genus 2 Riemann surface,
Kazhdan has constructed an $\varepsilon$-representation
$G{\>\longrightarrow\>}U(n)$, which cannot be $1/10$-aproximated by a
representation, for any given $\varepsilon>0$. In Subsection 5.3, we
generalize this result to all compact manifolds of negative sectional
curvature.

###  1.3 Quasimorphisms, bounded cohomology and the commutator length

In the literature, the quasimorphisms are usually considered as maps taking
values in ${\mathbb{R}}$. In this context, a quasimorphism is defined as a map
$q:\;G{\>\longrightarrow\>}{\mathbb{R}}$ which satisfies
$|q(xy)-q(x)-q(y)|<C$, where $C$ is a constant independent from $x,y$. We
shall sometimes call such maps commutative quasimorphisms.

This notion originates in the paper of Gromov [G2]. Using the quasimorphisms,
R. Brooks proved that the second bounded cohomology
$H^{2}_{b}({\mathbb{F}}_{2})$ of the free group ${\mathbb{F}}_{2}$ is
infinite-dimensional ([Br, G2]). Since then, quasimorphisms
$q:\;G{\>\longrightarrow\>}{\mathbb{R}}$ became prevalent in topology ([C]),
symplectic geometry ([EP, Sh]) and dynamics ([BM, GG]).

Barge and Ghys ([BG]) generalized the observation of Brooks to prove that the
second bounded cohomology of $\pi_{1}(S)$ is infinite-dimensional for any
Riemann surface $S$ with $g(S)>1$.

We say that two quasimorphisms $q,q_{1}:\;G{\>\longrightarrow\>}{\mathbb{R}}$
are equivalent if $|q(x)-q_{1}(x)|<C$, where $C$ is a constant independent
from $x$. The group ${\cal Q}$ of equivalence classes of quasimorphisms fits
into an exact sequence

$H^{1}(G,{\mathbb{R}}){\>\longrightarrow\>}{\cal
Q}{\>\longrightarrow\>}H^{2}_{b}(G,{\mathbb{R}}){\>\longrightarrow\>}H^{2}(G,{\mathbb{R}}),$

where $H^{2}_{b}(G,{\mathbb{R}})$ denotes the bounded cohomology. A
quasimorphism $q:\;G{\>\longrightarrow\>}{\mathbb{R}}$ is called homogeneous
if it satisfies $q(x^{n})=nq(x)$ for any $x\in G$ and $n\in{\mathbb{Z}}$. It
is easy to see that every quasimorphism
$q:\;G{\>\longrightarrow\>}{\mathbb{R}}$ is equivalent to a unique homogeneous
quasimorphism.

A group $G$ is called perfect if $G=[G,G]$. The commutator length of $g\in G$
is the minimal number $m$ such that $g$ can be represented as a product of $m$
commutators. Recall that the stable commutator length of an element $g\in G$
is defined as

$\mathop{\sf scl}(g):=\lim_{n\to\infty}\frac{\mathop{\sf cl}(g^{n})}{n},$

where $\mathop{\sf cl}$ is the commutator length. The homogeneous
quasimorphisms are related with the stable commutator length, due to the
celebrated theorem of Bavard ([B]). This result is known as Bavard duality. It
its simplest version, Bavard duality can be stated as follows: for any $g$ in
a perfect group $G$, $g$ has non-zero stable commutator length if and only if
there exists a homogeneous quasimorphism
$q:\;G{\>\longrightarrow\>}{\mathbb{R}}$ such that $q(g)\neq 0$.

###  1.4 Barge-Ghys construction and manifolds of negative curvature

Further on, we are interested in vector bundles over compact manifolds of
strictly negative sectional curvature. Sometimes such manifolds are called
“hyperbolic”, but we don’t use this term to avoid confusion with manifolds of
constant negative curvature.

We study the Ulam quasimorphisms associated with the holonomy of a connection
on vector bundles or on principal $G$-bundles, where $G$ is a connected Lie
group. In the sequel, we use either the language of $G$-bundles or the
language of vector bundles with connection, whatever is more convenient.
However, all statements that we make can be easily translated from one
language to another, giving equivalent results in two parallel frameworks.

Remark 1.2:  We tacitly assume that the base manifold of strictly negative
curvature has dimension at least 2.

Barge-Ghys quasimorphisms were introduced in [BG], who used them to prove that
the bounded cohomology of a Riemann surface is infinite-dimensional. Indeed,
the same argument works for any compact manifold of negative sectional
curvature. In the literature these quasimorphisms are variously called de Rham
quasimorphisms ([C]) or Barge-Ghys quasimorphisms [EP, RP]; we follow the
second convention.

###  1.5 Ulam stability and Kazhdan’s $\varepsilon$-representations

In another application of the non-commutative Barge-Ghys construction, we may
treat several questions on stability of homomorphisms, due to S. Ulam and D.
Kazhdan ([BOT], [U, VI.1], [Kaz]). Recall that a quasimorphism
$q:\;\Gamma{\>\longrightarrow\>}U(V)$ to is called an
$\varepsilon$-representation if $\|q(x)q(y)q(yx)^{-1}\|<\varepsilon$, where
$V$ a Hilbert space, which might be finite or infinite-dimensional, and
$\|\cdot\|$ the standard norm on $\operatorname{End}(V)$. We say that a
quasimorphism $q:\;\Gamma{\>\longrightarrow\>}U(V)$ is $\delta$-approximated
by a representation $\rho:\;\Gamma{\>\longrightarrow\>}U(V)$ if
$\|\rho(x)-q(x)\|<\delta$ for all $x\in\Gamma$.

The group $\Gamma$ is called Ulam stable ([BOT]) if for any $\delta>0$ there
exists $\varepsilon>0$ such that any finite-dimensional
$\varepsilon$-representation $q:\;\Gamma{\>\longrightarrow\>}U(V)$ can be
$\delta$-approximated by a representation
$\rho:\;\Gamma{\>\longrightarrow\>}U(V)$. It is called strongly Ulam stable if
the same is true even for infinite-dimensional Hilbert representations.

In [Kaz], D. Kazhdan has proven that all amenable groups are strongly Ulam
stable. He has also shown that for each $\varepsilon>0$ there exists an
$\varepsilon$-representation of $\pi_{1}(S)$, where $S$ is a compact Riemann
surface of genus 2, which cannot be $\frac{1}{10}$-approximated.

Using the Barge-Ghys quasimorphisms, we were able to prove this result for the
fundamental group of any closed negatively curved manifold. In 5.3, we
generalize [Kaz, Theorem 2] showing that a fundamental group $\Gamma$ of a
complete Riemannian manifold with uniformly bounded negative sectional
curvature admits an $\varepsilon$-representation taking values in any given
Lie group $G$, which cannot be $1/3$-approximated by a representation111Our
norm conventions are different from Kazhdan’s, but the constants $1/3$ and
$1/10$ depend on the choice of normalizations, and are not important in
themselves.. Kazhdan proves this for $\Gamma=\pi_{1}(S)$, where $S$ is a genus
2 Riemann surface, and $G=U(n)$.

###  1.6 Constructible quasimorphisms

The paper [FK] by Fujiwara and Kapovich is the first regular treatment of Ulam
quasimorphisms taking values in a non-commutative group $G$. Fujiwara and
Kapovich considered the case when $G$ is discrete, and their result is mostly
negative. Fujiwara and Kapovich defined “constructible quasimorphisms”, which
are essentially quasimorphisms which can be reduced to a composition of
homomorphisms and quasimorphisms taking values in an abelian group. Then they
proved that any Ulam quasimorphism taking values in a discrete group is always
constructible ([FK, Theorem 1.2]).

In 3.3 we construct examples of quasimorphisms taking values in a Lie group
$G$, which do not satisfy conclusions of [FK, Theorem 1.2]. There is no
contradiction because the group $G$ is not discrete. We prove that for any
compact manifold $M$ of strictly negative sectional curvature, there exists an
Ulam quasimorphism $\pi_{1}(M){\>\longrightarrow\>}G$ which is not
constructible.

##  2 Barge-Ghys quasimorphisms on fundamental groups of compact manifolds of
negative curvature.

###  2.1 Manifolds with negative sectional curvature

In this preliminary section we list a few geometric results, most of them
either classical or due to M. Gromov, [G2].

Theorem 2.1: (Cartan-Hadamard)
Let $M$ be a complete, simply connected Riemannian manifold of non-positive
sectional curvature. Then $M$ is contractible.

Proof: We give a sketch of the proof, following [BBI]. In [BBI] this result
was stated and proven for CAT(0)-spaces, but the comparison inequalities which
are required by the CAT(0)-geometry easily follow from the non-positivity of
the sectional curvature.

By Hopf-Rinow theorem, every two points of $M$ can be connected by a geodesic.
Let $\gamma_{1}:\;[a,b]{\>\longrightarrow\>}M$ and
$\gamma_{2}:\;[c,d]{\>\longrightarrow\>}M$ be segments of geodesics in $M$,
parametrized by the arc length. As shown in [BBI], the distance function
$D:\;[a,b]\times[c,d]{\>\longrightarrow\>}{\mathbb{R}}^{>0}$ taking $x,y$ to
$d(\gamma_{1}(x),\gamma_{2}(y))$ is strictly convex, unless the geodesics
$\gamma_{1},\gamma_{2}$ are segments of the same geodesic line. This implies,
in particular, that any two points are connected by a unique geodesic: indeed,
if $\gamma_{1}$ and $\gamma_{2}$ have the same ends,
$\gamma_{1}(a)=\gamma_{2}(c)$ and $\gamma_{1}(b)=\gamma_{2}(d)$ the function
$D$ would be equal to 0 in $(a,c)$ and $(b,d)$, hence it is zero on the
diagonal, and the images of $\gamma_{1}$ and $\gamma_{2}$ coincide.

Fix a reference point $p\in M$ and consider the function

$H:\;M\times[0,1]{\>\longrightarrow\>}M$

taking $x\in M$ and $t\in[0,1]$ to $\gamma(t\cdot d(p,x))$, where
$\gamma:\;[0,d(p,x)]$ is the geodesic connecting $p$ to $x$. A similar
argument implies that $H$ is continuous; clearly, $H$ is a deformation retract
of $M$ to $p$, hence $M$ is contractible.

We would not use the Cartan-Hadamard theorem, but we use its corollary, which
is inherent in its proof.

Corollary 2.2:  Let $M$ be a simply connected, complete manifold of non-
positive sectional curvature. Fix a reference point $p\in M$ and consider the
function $H_{p}:\;M\times[0,1]{\>\longrightarrow\>}M$ taking $x\in M$ and
$t\in[0,1]$ to $\gamma(t\cdot d(p,x))$, where
$\gamma:\;[0,d(p,x)]{\>\longrightarrow\>}M$ is the geodesic connecting $p$ to
$x$. Then $H_{p}$ is continuous.

We could use the map $H_{p}$ to define a straight singular simplex, following
[G2]. Since we need only 2-dimensional simplices, we restrict ourselves to the
2-dimensional case.

Definition 2.3: Let $M$ be a simply connected, complete manifold of non-
positive sectional curvature. Let $\gamma$ be the geodesic segment connecting
$b$ to $c$. The geodesic triangle $\Delta(a,b,c)$, associated with the points
$a,b,c\in M$ is the union $\cup_{t\in[0,1]}[H_{a}(t)(b),H_{a}(t)(c)]$, where
$H_{a}(t):\;M{\>\longrightarrow\>}M$ is the homotopy along geodesics passing
through $a$, defined in 2.1, and $[H_{a}(t)(b),H_{a}(t)(c)]$ the geodesic
segment connecting $H_{a}(t)(b)$ and $H_{a}(t)(c)$.

Remark 2.4: The boundary of the triangle $\Delta(a,b,c)$ is the union of
geodesics connecting $a$ to $b$, $b$ to $c$ and $c$ to $a$. Indeed, the
homothety $H_{a}(t)$ moves any point $x\in M$ along the geodesic connecting
$x$ to $a$ as $t$ goes from $1$ to $0$, hence the segments of the boundary
connecting $b$ to $a$ and $c$ to $a$ are geodesics; the third segment is
$\gamma$, which is also chosen geodesic.

Remark 2.5: Note that the order of the points $a,b,c\in M$ is important.
Indeed, unless $M$ has constant sectional curvature, the triangles
$\Delta(a,b,c)$ and (say) $\Delta(b,a,c)$ are different: otherwise, if
$\Delta(a,b,c)=\Delta(b,a,c)=\Delta(a,c,b)$, this triangle is a segment of a
completely geodesic 2-dimensional submanifod, and a general Riemannian
manifold does not have completely geodesic submanifolds ([MW]).

The main technical result about manifolds of negative sectional curvature
which we use is the following theorem of Gromov, which is proven in [G2] for
any straight singular simplex.

Theorem 2.6:  Let $M$ be a simply connected, complete manifold of strictly
negative sectional curvature $K(M)<-\varepsilon<0$, and $\Delta(a,b,c)$ a
geodesic triangle defined above. Then the Riemannian area
$\operatorname{\text{\it Area}}(\Delta(a,b,c))$ satisfies
$\operatorname{\text{\it Area}}(\Delta(a,b,c))\leqslant\pi\varepsilon^{-2}$.

Proof: [G2, §I.3].

###  2.2 Non-commutative Barge-Ghys quasimorphisms

Definition 2.7:  Let $M$ be a compact manifold with non-positive sectional
curvature, and $(P,\nabla)$ a principal $G$-bundle with connection. Fix $x\in
M$. The non-commutative Barge-Ghys map takes $\gamma\in\pi_{1}(M,x)$ to the
holonomy of $\nabla$ along the geodesic path starting and ending at $x$ and
homotopic to $\gamma$.111Since $M$ has non-positive curvature, this geodesic
path is unique in its homotopy class.

Remark 2.8: Throughout this paper, we consider the maps
$q:\;\Gamma{\>\longrightarrow\>}G$, where $\Gamma=\pi_{1}(M)$ is the
fundamental group of a compact manifold of strictly negative curvature. Such
groups are Gromov hyperbolic, however, it is not clear yet whether our
constructions can be generalized to all Gromov hyperbolic groups.
Nevertheless, all results we prove remain valid whenever each geodesic
triangle in the universal cover of $M$ can be filled by a disk of bounded
area, as in 2.1.

Theorem 2.9:  Let $M$ be a compact manifold of negative sectional curvature,
and $\Theta$ a geodesic $n$-polygon in $M$, that is, a contractible loop of
$n$ geodesic segments. Consider a principal $G$-bundle $(P,\nabla)$ with
connection on $M$, and let $h(\Theta)\in G$ be the holonomy along the boundary
of $\Theta$, considered as a loop starting and ending at $p\in\Theta$. Then
$h(\Theta)$ belongs to a compact $K_{n}\subset G$ which is independent from
the choice of $\Theta$, but depends on the choice of $n$ and $(P,\nabla)$ and
the bound on the curvature of $M$.

Proof: By the effective version of the Ambrose-Singer theorem, the holonomy
along a path is bounded by the integral of the curvature over a disk filling
this path ([RW, Theorem 1], [MO1], [Y]). The area of any geodesic triangle is
bounded by 2.1. The absolute value of the curvature of $\nabla$ is bounded
from above because $M$ is compact, and the curvature form on
$(\widetilde{P},\widetilde{\nabla})$ is obtained as a pullback of the
curvature of $(P,\nabla)$.

This implies that $h(\Theta)$ belongs to a fixed compact $K$ when $n=3$ and
$\Theta$ is a triangle. When $n>3$, we represent $\Theta$ as a boundary of the
union of $n-2$ geodesic triangles $D_{1},..,D_{n-2}$ with common vertex $p$.
Then the holonomy $h(\Theta)$ is obtained as a product
$h(\Theta)=h(D_{1})h(D_{2})...h(D_{n-2})\subset K^{n-2}$. Therefore,
$h(\Theta)$ belongs to a fixed compact $K_{n}:=K^{n-2}$, independent from the
choice of $\Theta$.

Theorem 2.10:  Let $M$ be a compact manifold of negative sectional curvature,
and $(P,\nabla)$ a principal $G$-bundle with connection. Fix $x\in M$, and let
$q:\;\pi_{1}(M){\>\longrightarrow\>}G$ be the non-commutative Barge-Ghys map
associated with $(P,\nabla)$. Then $q$ is an Ulam quasimorphism (1.1).

Proof: Denote by
$\widetilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$ the
universal cover of $M$. Let $a,b\in\pi_{1}(M)$, and $P_{a}$,
$P_{b}\in\operatorname{Diff}(\widetilde{M})$ the corresponding deck
transformations. Fix a preimage $\widetilde{x}\in\pi^{-1}(x)$, and denote by
$(\widetilde{P},\widetilde{\nabla})$ the pullback of $(P,\nabla)$ to
$\widetilde{M}$. By definition, the product $q(ab)q(b)^{-1}q(a)^{-1}$ is
represented by the holonomy of $(\widetilde{P},\widetilde{\nabla})$ along the
geodesic triangle connecting the points $\widetilde{x},P_{a}(\widetilde{x}),$
and $P_{b}(P_{a}(\widetilde{x}))$ in $\widetilde{M}$. By 2.2, this quantity
belongs to a compact subset independent from the choice of $x\in M$ and
$a,b\in\pi_{1}(M)$.

Remark 2.11: The set of equivalence classes of non-commutative Barge-Ghys
quasimorphisms is very big. As follows 2.3 below, the usual (“commutative”)
Barge-Ghys quasimorphism is a special case of the quasimorphism taking values
in a Lie group as defined in 2.2. By [BG], [C], the vector space spanned by
commutative Barge-Ghys quasimorphisms up to equivalence is infinite-
dimensional; in 3.3 we construct non-commutative Barge-Ghys quasimorphisms
which cannot be reduced to commutative ones.

Remark 2.12: Using the Riemann-Hilbert correspondence, one can associate a
flat principal $G$-bundle over $M$ with each group homomorphism
$\rho:\;\pi_{1}(M){\>\longrightarrow\>}G$. By definition, the corresponding
Barge-Ghys quasimorphism is equal to $\rho$. In other words, all group
homomorphisms to a Lie group can be realized as Barge-Ghys quasimorphisms.

###  2.3 Commutative and non-commutative Barge-Ghys quasimorphisms

Definition 2.13:  Let $M$ be a compact manifold of strictly negative sectional
curvature, and $\theta\in\Lambda^{1}M$ a 1-form on $M$. The (commutative)
Barge-Ghys quasimorphism ([BG]) associated with $\theta$ takes
$\gamma\in\pi_{1}(M)$ to the integral of $\theta$ over the geodesic path
starting and ending at $x$ and homotopic to $\gamma$.

Let $B$ be an oriented real rank-1 vector bundle on $M$. Topologically, $B$ is
always trivial. To trivialize $B$, we choose a metric on $B$ using the
partition of unity, and trivialize $B$ by taking a positive length-1 section
$u$. Fix the connection $\nabla_{0}$ on $B$ in such a way that
$\nabla_{0}u=0$. Then any connection $\nabla$ on $B$ can be written as
$\nabla=\nabla_{0}+\theta$, where $\theta\in\Lambda^{1}(M)$ is a 1-form. The
holonomy of this connection along a loop $\gamma$ is given by
$\operatorname{Hol}_{\gamma}(\nabla)=e^{-\int_{\gamma}\theta}$. Indeed, for
any section $u_{f}:=f(t)u$ of $B$ restricted to a geodesic segment $\gamma$
parametrized by $t\in[a,b]$, the equation

$\nabla(u_{f})=\frac{df}{dt}u+\theta fu=0$

is equivalent to $f^{\prime}=-f\theta$, equivalently, $\frac{d\log
f}{dt}=-\theta$.

Proposition 2.14:  Let $(B,\nabla_{0})$ be a trivial real rank 1 vector bundle
on a compact manifold $M$ of negative sectional curvature $\theta$ a 1-form on
$M$ and $\nabla:=\nabla_{0}-\theta$. Then the commutative Barge-Ghys
quasimorphism associated with $\theta$ can be obtained as the logarithm of the
non-abelian Barge-Ghys quasimorphism associated with $(B,\nabla)$.

Proof: Indeed, for any loop in $M$, one has
$\operatorname{Hol}_{\gamma}(\nabla)=e^{-\int_{\gamma}\theta}$.

Remark 2.15: For any trivialized real rank 1 vector bundle $B$ on $M$, the
connections on $B$ are in bijective correspondence with 1-forms on $M$.
Therefore, 2.3 defines a bijective correspondence between the set of (non-
commutative) Barge-Ghys quasimorphisms associated with $B$ and the set of
commutative Barge-Ghys quasimorphisms.

##  3 Constructible Ulam quasimorphisms

In the sequel, when we speak of nilpotent Lie groups, we always mean
connected, simply connected, algebraic nilpotent Lie groups over
${\mathbb{R}}$.

###  3.1 Zariski closure of discrete subgroups

Definition 3.1: Let $Y$ be a real algebraic variety, and $X\subset Y$ a
subset. Zariski closure of $X$ in $Y$ is the intersection of all real
algebraic subvarieties $X_{i}\subset Y$ containing $X$. A subset $X\subset Y$
is Zariski dense if its Zariski closure is $Y$.

Remark 3.2: Let $\Gamma\subset G$ be a subgroup of a real algebraic group.
Then its Zariski closure $\overline{\Gamma}$ is an algebraic subgroup of $G$.
Indeed, the group laws put algebraic constraints on $\overline{\Gamma}$, hence
the smallest algebraic subvariety of $G$ containing $\Gamma$ is closed under
the group operations.

The following notion is going to be used in the proof of 3.3 below. For the
readers convenience, we recall the definition of Hausdorff distance ([G1]).

Definition 3.3:  Let $Z_{1},Z_{2}$ be subsets of a metric space $M$. Denote by
$Z_{i}(\varepsilon)$ the $\varepsilon$-neighbourhood of $Z_{i}$, and let

$d_{H}(Z_{1},Z_{2}):=\inf\\{\varepsilon\in{\mathbb{R}}^{>0}\cup\infty\ \ |\ \
Z_{1}(\varepsilon)\supset Z_{2}\text{\ and \ }Z_{2}(\varepsilon)\supset
Z_{1}\\}.$

The number $d_{H}(Z_{1},Z_{2})$ is called the Hausdorff distance between
$Z_{1}$ and $Z_{2}$. It is not hard to see that $d_{H}$ defines a metric,
taking values in ${\mathbb{R}}^{>0}\cup\infty$, on the set of all closed
subsets of $M$.

Definition 3.4: Two subsets $X,Y$ of a metric space $M$ are coarse equivalent
if the Hausdorff distance $d_{H}(X,Y)$ is finite. When $M$ is a Lie group, and
$d$ a left-invariant Riemannian distance, this is equivalent to $K\cdot
X\supset Y$ and $K\cdot Y\supset X$ for a compact subset $K\subset G$.

Example 3.5: Any two rank 2 discrete lattices in ${\mathbb{R}}^{2}$ with the
usual Euclidean metric are coarse equivalent. However, ${\mathbb{R}}^{2}$ is
not coarse equivalent to a point.

We need to relax the notion of coarse equivalence slightly to accomodate the
metrics which are not bi-invariant.

Definition 3.6: Let $G$ be a Lie group. We say that $X\subset G$ is bi-coarse
equivalent to $Y\subset G$ if there exists a compact subset $K\subset G$ such
that $K\cdot X\cdot K\supset Y$ and $K\cdot Y\cdot K\supset X$.

Definition 3.7:  Let $G$ be a real algebraic Lie group, and $\Lambda\subset G$
a Zariski dense subgroup. We say that $\Lambda$ is bi-coarse Zariski dense in
$G$ if the following property holds. If $\Lambda^{\prime}\subset G$ is bi-
coarse equivalent to $\Lambda$, then $\Lambda^{\prime}$ is also Zariski dense
in $G$.

We prove that a lattice in a nilpotent Lie group is bi-coarse Zariski dense.
Note that simply connected nilpotent Lie groups are algebraic, with the
algebraic structure induced by the standard algebraic structure on its Lie
algebra. Indeed, for a nilpotent Lie group, the exponential map from its Lie
algebra to the Lie group is polynomial and invertible, and the inverse map,
called the logarithm, is also polynomial ([CG, Proposition 1.2.8]).

Definition 3.8: A subset $\Lambda\subset G$ of a Lie group is called bi-
cocompact if there exists a compact subset $K\subset G$ such that
$K\cdot\Lambda\cdot K=G$.

Further on, we use the following elementary claim. Clearly, the subgroup
$[G,G]\subset G$ is normal, and for any $H\subset G$ the group $H\cdot[G,G]$
generated by $H$ and $[G,G]$ is also normal.

Claim 3.9:  Let $G$ be a connected, simply connected algebraic nilpotent Lie
group, and $H\subset G$ a proper algebraic subgroup. Then
$G/(H\cdot[G,G])\cong{\mathbb{R}}^{n}$, where $n>0$.

Proof: A quotient of a connected, simply connected nilpotent Lie group by an
algebraic subgroup is connected, simply connected, because this algebraic
subgroup is connected and simply connected ([CG, Proposition 1.2.8]). Since
$G/(H\cdot[G,G])$ is commutative, it is isomorphic to ${\mathbb{R}}^{n}$. It
remains to show that $n>0$. Otherwise the projection of $\mathfrak{h}$ to
$\frac{{\mathfrak{g}}}{[{\mathfrak{g}},{\mathfrak{g}}]}$ is surjective, where
${\mathfrak{g}}=\operatorname{Lie}(G)$ and
$\mathfrak{h}=\operatorname{Lie}(H)$. However, any set of elements generating
$\frac{{\mathfrak{g}}}{[{\mathfrak{g}},{\mathfrak{g}}]}$ also generates
${\mathfrak{g}}$ ([Ko]), hence in this case $G=H$, which is a contradiction
because $H\subset G$ is a proper subgroup.

Remark 3.10: Let $H\subset G$ be a Zariski closed subgroup in a connected,
simply connected nilpotent algebraic Lie group. Then $H$ is also connected and
simply connected ([CG, Proposition 1.2.2]).

Lemma 3.11:  Let $\Lambda\subset G$ be a bi-cocompact subset in a simply
connected nilpotent Lie group $G$. Then the group generated by $\Lambda$ is
Zariski dense.

Proof: Let $H\subset G$ be the Zariski closure of the group generated by
$\Lambda$. Since $H$ is bi-cocompact, $K\cdot H\cdot K=G$ for some compact
$K\subset G$. Consider the projection map
$\pi:\;G{\>\longrightarrow\>}G/[G,G]$. Let $K_{1}:=\pi(K)$; this set is
compact because $K$ is compact and $\pi$ is continuous. Then
$K_{1}\pi(H)K_{1}=\pi(G)$, which is impossible, because
$\pi(H)={\mathbb{R}}^{k}\subsetneq\pi(G)={\mathbb{R}}^{n}$ (3.1).

Corollary 3.12:  Let $G$ be a simply connected nilpotent Lie group, and
$\Gamma\subset G$ a lattice, which is cocompact ([W, Theorem 2.2.6]). Then
$\Gamma$ is bi-coarse Zariski dense.

Proof: Since $\Gamma$ is cocompact, it is bi-cocompact. Then it is bi-coarse
Zariski dense by 3.1.

###  3.2 Infinitely many cyclic subgroups in Gromov hyperbolic groups

In the proof of 3.3 below we need the following statement about hyperbolic
groups, which is essentially due to Epstein and Fujiwara, [EF, Proposition
3.6].

In the sequel, we use $X^{u}$ to denote the conjugation by $u$,
$X^{u}=uXu^{-1}$. A group is called virtually cyclic if it contains a cyclic
group of finite index. A quasimorphism
$q:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}$ is called homogeneous if
$q(x^{n})=nx$ for all $n\in{\mathbb{Z}}$ and all $x\in\Gamma$.

Claim 3.13:  Let $\Gamma$ be an infinite Gromov hyperbolic group which is not
virtually cyclic. Then $\Gamma$ contains infinitely many cyclic subgroups
$U_{i}\subset\Gamma$ which satisfy the following condition

$\forall u,v\in\Gamma,\forall i\neq j,\text{\ one has\ }U_{i}^{u}\cap
U_{j}^{v}=1.$ (3.1)

Proof: By [EF, Proposition 3.6], there exists an infinite family of elements
$w_{i}\in\Gamma$ and an infinite family of homogeneous quasimorphisms
$h_{i}:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}$ such that $h_{i}(w_{j})=0$
for all $i<j$ and $h_{i}(w_{i})=1$. It is easy to see that all homogeneous
quasimorphisms are invariant under conjugation ([C, §2.2.3]). It follows that
for any $i<j$ and all $k,l\neq 0$, the element $w_{i}^{k}$ is not conjugated
to $w_{j}^{l}$. Then the cyclic subgroups $V_{i}=\langle w_{i}\rangle$ satisfy
(3.1).

The following corollary is probably very well known, but we did not find a
more economical way to prove it.

Corollary 3.14:  Let $\Gamma$ be an infinite Gromov hyperbolic group. Then the
number of conjugacy classes of elements in $\Gamma$ is infinite.

Proof: 3.2 for a virtually cyclic group is trivial and left as an exercise.
When $\Gamma$ is not virtually cyclic, we apply 3.2 and find an infinite
number of elements $w_{i}$ of $\Gamma$ generating $V_{i}$ and pairwise non-
conjugate.

The following claim is especially useful in conjunction with [EF, Proposition
3.6].

Claim 3.15:  Let $a,b\in\Gamma$ be elements of a Gromov hyperbolic group
$\Gamma$. Assume that there exist homogeneous quasimorphisms
$q,s:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}$ such that
$q(a)=0,q(b)=1,s(a)=1,s(b)=0$. Then for almost all $k\neq 0,l\neq 0$, the
elements $a^{k}b^{l}$ are not conjugate to powers of $a$ and $b$.

Proof: Let $D(q)$ be the defect of the quasimorphism $q$, defined as
$D(q)=\sup_{x,y\in\Gamma}|q(xy)-q(x)-q(y)|$. Suppose that $a^{k}b^{l}$ is
conjugate to $a^{m(k,l)}$, for infinitely many $l$. Then

$0=q(a^{m(k,l)})=q(a^{k}b^{l})=q(a^{k})+q(b^{l})+\lambda=lq(b)+\lambda,$

where $|\lambda|\leqslant D(q)$. This might happen only if $q(b)=0$,
contradicting the assumption $q(b)=1$.

###  3.3 Non-constructible Ulam quasimorphisms

In [FK], Kapovich and Fujiwara define a constructible quasimorphism taking
values in a discrete group. We extend their definition to any Lie group.

Definition 3.16:  Let $q:\;\Gamma{\>\longrightarrow\>}G$ be an Ulam
quasimorphism. Consider a finite index subgroup $\Gamma_{0}\subset\Gamma$, and
let $H\supset q(\Gamma_{0})$ be a subgroup of $G$ containing $q(\Gamma_{0})$.
Assume that $H$ contains a normal subgroup $A$, abelian and central in $H$,
such that the composition $\Gamma_{0}\stackrel{{\scriptstyle
q}}{{{\>\longrightarrow\>}}}H{\>\longrightarrow\>}H/A$ is equivalent to a
homomorphism $\Gamma_{0}{\>\longrightarrow\>}H/A$. Then $q$ is called a
constructible quasimorphism.

An element $a\in\Gamma$ of a group is called primitive if for any
$n\in{\mathbb{Z}}$ and $b\in\Gamma$ such that $b^{n}=a$, one has $n=\pm 1$. A
quasimorphism $q:\;\Gamma{\>\longrightarrow\>}G$ is called homogeneous if its
restriction to any cyclic subgroup is a homomorphism (4.2). We give a detailed
treatment of homogeneous quasimorphisms in Section 4, where we define a new
type of quasimorphisms, called HBG-quasimorphism, for “homogeneous Barge-
Ghys”. We use the acronym HBG to avoid the confusion, because the HBG-
quasimorphisms are not, in fact, “Barge-Ghys quasimorphisms”, in the sense of
2.2. One should think of a HBG-quasimorphism as of a “homogenization” of a
Barge-Ghys quasimorphism. However, the notion of homogenization, which is well
known for the usual (${\mathbb{R}}$-valued) quasimorphisms, is not defined
(yet) for the quasimorphisms taking values in a Lie group.

In Section 4, prove the following theorem.

Theorem 3.17:  Let $M$ be a compact manifold of strictly negative curvature,
$G$ a connected non-abelian Lie group, and
$x_{1},...,x_{n}\in\Gamma:=\pi_{1}(M)$ a collection of primitive elements
generating cyclic subgroups $U_{i}$ satisfying

$\forall u,v\in\Gamma,\forall i\neq j,\text{\ one has\ }U_{i}^{u}\cap
U_{j}^{v}=1$ (3.2)

where $U_{i}^{u}$ denotes the $u$-conjugate of $U_{i}$.111Since $\dim M>1$,
there exist infinitely many such subgroups, as follows from 3.2. Take any
collection of elements $g_{i}\in G$, with $i=1,...,n$. Then

(i)

There exists a connection $\nabla$ on a trivial principal bundle $P$ such that
the corresponding HBG-quasimorphism
$q_{\nabla}:\;\Gamma{\>\longrightarrow\>}G$ takes $x_{i}$ to $g_{i}$,
$i=1,2,...,n$.

(ii)

Moreover, for any countable family of cyclic subgroups $V_{i}\subset\Gamma$
satisfying

$\forall u,v\in\Gamma,\forall i\neq j,\text{\ one has\ }V_{i}^{u}\cap
U_{j}^{v}=1$

and $q_{\nabla}(V_{i})$ is not central in $G$.

Proof: See Subsection 5.2 below.

We use this result to prove the following theorem.

Theorem 3.18:  Let $G$ be a simply connected, connected, non-abelian rational
real nilpotent Lie group,222A real nilpotent Lie group is called rational if
its Lie algebra ${\mathfrak{g}}$ contains a rational Lie subalgebra
${\mathfrak{g}}_{\mathbb{Q}}$ such that
${\mathfrak{g}}={\mathfrak{g}}_{\mathbb{Q}}\otimes_{\mathbb{Q}}{\mathbb{R}}$.
By Maltsev’s theorem, this property is equivalent to existence of cocompact
lattices ([CG]).  and $\Gamma:=\pi_{1}(M)$, where $M$ is a compact manifold of
strictly negative sectional curvature. Then there exists a non-constructible
HBG-quasimorphism $q_{\nabla}:\;\Gamma{\>\longrightarrow\>}G$.

Proof. Step 1: By Maltsev’s theorem, $G$ contains a lattice $\Lambda$ ([CG,
Theorem 5.8.1]), which is a posteriori cocompact ([CG, Corollary 5.4.6]). This
lattice is finitely generated ([CG, Corollary 5.1.7]). Denote its generators
by $g_{i}$. By definition, $\Gamma$ is a fundamental group of a hyperbolic
manifold $M$ (that is, a compact manifold of strictly negative sectional
curvature), and $\dim_{\mathbb{R}}M>1$. By 3.2, $\Gamma$ contains infinitely
many free cyclic subgroups satisfying (5.1). Using 3.3, we find
$x_{1},...,x_{n}\in\Gamma$ and a HBG-quasimorphism
$q_{\nabla}:\;\Gamma{\>\longrightarrow\>}G$ taking each $x_{i}$ to $g_{i}$. We
choose, in addition, elements $a,b\in\Gamma$ such that $a^{l}b^{m}$ are not
conjugate to powers of $a,b$ for almost all $l,m\in{\mathbb{Z}}$ (3.2). Using
an appropriate family of quasimorphisms as in the proof of 3.2, we may choose
$a$ and $b$ in such a way that the cyclic subgroups $V_{i}$ generated by the
elements $x_{1},...,x_{n},a,b$ satisfy (5.1).

We apply 3.3 to $V_{i}$ as above, $U_{i}$ generated by those $a^{l}b^{m}$
which are not conjugate to the powers of $a,b$. Then we obtain a HBG-
quasimorphism such that $q_{\nabla}(x_{i})=g_{i}$, $q_{\nabla}(a)=1$,
$q_{\nabla}(b)=1$, and $q_{\nabla}(a^{i}b^{j})$ is not central for almost all
$i,j>0$.

We are going to show that $q_{\nabla}$ is not constructible.

Step 2: By contradiction, suppose that $q_{\nabla}$ is constructible, that is,
there exists a finite index subgroup $\Gamma_{0}\subset\Gamma$ and
$q_{\nabla}{\left|{}_{{\phantom{|}\\!\\!}_{\Gamma_{0}}}\right.}$ satisfies the
assumptions of 3.3.

Since $\Gamma_{0}$ is of finite index in $\Gamma$, it contains powers
$x_{i}^{n_{i}}$ of each $x_{i}$, for some $n_{i}\in{\mathbb{Z}}^{>0}$. Using
the central series for the lattice $\Lambda\subset G$, we can easily see that
the elements $q_{\nabla}(x_{i}^{n_{i}})=g_{i}^{n_{i}}$ generate a finite index
sublattice $\Lambda_{0}\subset\Lambda$, which is bi-coarse equivalent to $G$
because it is cocompact.

Since $\Lambda_{0}$ is bi-coarse Zariski dense (3.1), any map
$\Gamma_{0}{\>\longrightarrow\>}G$ which is equivalent to $q_{\nabla}$ has a
Zariski dense image.

Since $q_{\nabla}$ is constructible, there exists a homomorphism
$q:\;\Gamma_{0}{\>\longrightarrow\>}H/A$, where $H\subset G$ is a subgroup and
$A$ its central subgroup, which can be lifted to a map
$\Gamma_{0}{\>\longrightarrow\>}H$ coarse equivalent to
$q_{\nabla}{\left|{}_{{\phantom{|}\\!\\!}_{{\Gamma_{0}}}}\right.}$. Then
$\operatorname{im}q$ is Zariski dense in $G$, as indicated above. Therefore
the group $H$ is also Zariski dense in $G$, and $A$ belongs to the center
$Z\subset G$.

Clearly, the composition of $q:\;\Gamma_{0}{\>\longrightarrow\>}G/A$ and
$\pi:\;G/A{\>\longrightarrow\>}G/Z$ is a homomorphism. It remains to show that
the composition of
$q_{\nabla}{\left|{}_{{\phantom{|}\\!\\!}_{\Gamma_{0}}}\right.}$ and
$\pi:\;G{\>\longrightarrow\>}G/Z$ is not equivalent to any homomorphism, which
will give a contradiction.

Step 3: A sequence $\\{x_{i}\in G\\}$ is called bounded if it belongs to a
compact subset of $G$. For any compact subset $K\subset G$ in a topological
group $G$, and any sequences $\\{x_{i}\\}$, $\\{y_{i}\in Kx_{i}K\\}$, the
sequence $\\{x_{i}\\}$ is bounded if and only if $\\{y_{i}\\}$ is bounded.
Indeed, $y_{i}\in Kx_{i}K$ $\Leftrightarrow$ $x_{i}\in K^{-1}y_{i}K^{-1}$, and
for any bounded $\\{x_{i}\in K_{1}\\}$, the sequence $y_{i}\in Kx_{i}K\subset
KK_{1}K$ is clearly bounded.

Step 4: Let $q_{1}:\;\Gamma{\>\longrightarrow\>}G/Z$ denote the composition of
$q_{\nabla}:\;\Gamma{\>\longrightarrow\>}G$ and the projection to $G/Z$.
Arguing by contradiction, suppose that $q:\;\Gamma{\>\longrightarrow\>}G/Z$ is
a homomorphism which is equivalent to $q_{1}$. Then there exists a compact
$K\subset G$ such that

$q(z)=Kq_{1}(z)K$ (3.3)

for all $z\in\Gamma_{0}$. Since $\Gamma_{0}$ has finite index in $\Gamma$,
there exists $i,j>0$ such that $a^{i},b^{j}\in\Gamma_{0}$. Since
$q_{1}(a^{ki})=q_{1}(b^{kj})=1$ for all $k\in{\mathbb{Z}}$, this gives
$q(a^{ik})=q(a^{i})^{k}\in K\cdot K$ and $q(b^{jk})=q(b^{j})^{k}\in K\cdot K$.
However, for any element $x$ in a simply connected nilpotent group, a sequence
$\\{x^{n},n\in{\mathbb{Z}}\\}$ can be bounded only if $x=1$, hence
$q(a^{i})=1$ and $q(b^{j})=1$. Since $q$ is a homomorphism, this implies that
$q(a^{i}b^{j})=1$. Since $q_{1}$ is homogeneous, and $q_{1}(a^{i}b^{j})\neq
1$, the sequence $q_{1}((a^{i}b^{j})^{n}),n\in{\mathbb{Z}}$ is not bounded,
which implies that the sequence $q((a^{i}b^{j})^{n})\in
Kq_{1}((a^{i}b^{j})^{n})K$ is also unbounded (Step 3), giving a contradiction.

Remark 3.19:  In 3.3, we construct a HBG-quasimorphism which is not equivalent
to a constructible one. However, the proof we use brings a more powerful
result, which was the chief aim of an earlier version of this paper. Let
$q,q^{\prime}:\;\Gamma{\>\longrightarrow\>}G$ be Ulam quasimorphisms. We say
that $q,q^{\prime}$ are algebraically equivalent if there exists a compact
subset $K\subset G$ such that for all $x\in\Gamma$, one has $q(x)\in
Kq^{\prime}(x)K$. In 3.3, we construct a HBG-quasimorphism $q_{\nabla}$ which
is not algebraically equivalent to a constructible quasimorphism.

###  3.4 Different notions of quasimorphisms: geometric, algebraic and Ulam

The notion of Ulam quasimorphism is not the only notion of a quasimorphism
considered in the literature. In [HS], Hartnick and Schweizer define an
alternative notion of “a quasimorphism” $\varphi:\;G{\>\longrightarrow\>}H$ as
a map such that for any quasimorphism $q:\;H{\>\longrightarrow\>}{\mathbb{R}}$
its composition with $\varphi$ is a quasimorphism
$q\circ\varphi:\;G{\>\longrightarrow\>}{\mathbb{R}}$. When $H$ is a hyperbolic
group, it has many quasimorphisms to ${\mathbb{R}}$ (3.2), and this condition
is quite restrictive. However, when $H$ is a lattice of high rank, such as
$SL(n,{\mathbb{Z}})$, $n>2$, ([BM1, BM2]), all quasimorphisms to
${\mathbb{R}}$ are equivalent to homomorphisms; for such $H$ the Hartnick-
Schweizer condition is quite weak.

In [FK], Fujiwara and Kapovich considered other versions of the notion of
quasimorphism, taking values in a non-commutative group $H$; when $H$ is
commutative, all these notions are equivalent to the notion of Ulam
quasimorphism. They define the algebraic quasimorphisms as maps
$q:\;G{\>\longrightarrow\>}H$ such that there exists a compact set $K$ and
$q(xy)\in Kq(x)Kq(y)K$ for all $x,y\in G$, and geometric quasimorphisms as
ones such that $q(xy)\in Kq(x)Kq(y)$.

For comparison, Ulam quasimorphisms are maps $q:\;G{\>\longrightarrow\>}H$
which satisfy $q(xy)\in Kq(x)q(y)$; this notion is stronger than the notion of
a geometric quasimorphism.

Since $q(x^{-1})=q(x)^{-1}$, the condition $q(xy)\in Kq(x)q(y)$ is equivalent
to $q(xy)\in q(x)q(y)K$. Indeed, $q((xy)^{-1})\in Kq(y^{-1})q(x^{-1})$ implies
$q(xy)\in(Kq(y^{-1})q(x^{-1}))^{-1}=q(x)q(y)K^{-1}$. The same argument works
for the geometric quasimorphisms if we also assume that $q(x^{-1})=q(x)^{-1}$.

Earlier, we defined an equivalence of quasimorphisms: $q\sim q^{\prime}$ if
there exists a compact $K\subset H$ such that $q(x)\in Kq^{\prime}(x)$ for all
$x\in G$. Interestingly enough, a map which is equivalent to an Ulam
quasimorphism is no longer an Ulam quasimorphism (see 3.4 below). However, a
map $q^{\prime}:\;G{\>\longrightarrow\>}H$ which is equivalent to a geometric
quasimorphism $q:\;G{\>\longrightarrow\>}H$ is again a geometric
quasimorphism:

$\displaystyle q(xy)$ $\displaystyle\in Kq(x)Kq(y)\Leftrightarrow$
$\displaystyle q^{\prime}(xy)$ $\displaystyle\in K_{1}q(xy)\subset
K_{1}Kq(x)Kq(y)\subset K_{1}KK^{-1}_{1}q^{\prime}(x)KK^{-1}_{1}q^{\prime}(y).$

Here the “equivalence” is understood as $q^{\prime}(x)\in K_{1}q(x)$, and the
“geometric quasimorphism” condition as $q(xy)\in Kq(x)Kq(y)$.

The reason why the “geometric quasimorphism” condition is sometimes more
appropriate stems from this observation. Let $q:\;G{\>\longrightarrow\>}H$ be
an Ulam quasimorphism, and $d:\;H\times
H{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$ a left-invariant metric such
that all closed balls in $H$ are compact. Then the condition
$d(q(xy),q(x)q(y))\leqslant C$ is equivalent to being an Ulam quasimorphism.
However, a map $q^{\prime}:\;G{\>\longrightarrow\>}H$ such that
$d(q(x),q^{\prime}(x))\leqslant C_{1}$ is no longer an Ulam quasimorphism, but
only a geometric one. This leads to the following observation.

Remark 3.20:  Let $q:\;\Gamma{\>\longrightarrow\>}G$ be a non-constructible
quasimorphism (3.3) taking values in a nilpotent Lie group $G$ containing a
cocompact lattice $\Lambda$. Pick a right-invariant metric $d$ on $G$, and let
$R$ be the diameter of the fundamental domain of $\Lambda$ acting on $G$. For
each $q(x)\in G$, let us choose a closest element $q^{\prime}(x)\in\Lambda$.
Then $d(q(x),q^{\prime}(x))<R$, hence the map
$q^{\prime}:\;\Gamma{\>\longrightarrow\>}\Lambda$ is equivalent to $q$. This
gives an example of a geometric quasimorphism which is non-constructible, even
when the target group is discrete. By [FK], any Ulam quasimorphism
$q^{\prime\prime}:\;\Gamma{\>\longrightarrow\>}\Lambda$ into a discrete group
is constructible, hence $q^{\prime}$ is not equivalent to any Ulam
quasimorphism.

##  4 HBG-quasimorphisms

###  4.1 Translation length in Gromov hyperbolic groups

Let $\Gamma$ be a group generated by a finite set ${\cal A}$, and $C_{\cal
A}(\Gamma)$ its Cayley graph. Algebraic translation length $\tau(\gamma)$ of
an element $\gamma\in\Gamma$ is defined ([BH]) as

$\tau(\gamma):=\lim_{n\to\infty}\frac{1}{n}d(1,\gamma^{n}).$

The limit exists because the function $n\mapsto d(1,\gamma^{n})$ is
subadditive. It is not hard to see that the map
$\gamma{\>\longrightarrow\>}\tau(\gamma)$ is conjugation invariant ([BH,
Remark $\Gamma$.3.14 (1)]). Further on, we shall use the following result,
also found in [BH].

Proposition 4.1:  Let $\Gamma$ be a finitely generated Gromov hyperbolic
group, and $S_{R}$ the set of all conjugacy classes of all $\gamma\in\Gamma$
satisfying $\tau(\gamma)<R$, where $R>0$ is a real number. Then $S_{R}$ is
finite.

Proof: [BH, Proposition $\Gamma$.3.15].

An element $\gamma\in\Gamma$ is called primitive if it cannot be represented
as a power $\gamma=\varphi^{n}$, for any $n>1$.

Corollary 4.2:  Let $\Gamma$ be a finitely generated Gromov hyperbolic group.
Then any non-torsion element of $\Gamma$ is an integer power of a primitive
element.

Proof: By [C, Corollary 3.3.5], $\tau(\gamma)>0$ for all non-torsion $\gamma$.
Clearly, $\tau(\gamma^{k})=k\tau(\gamma)$. By 4.1, there exists a number
$C\in{\mathbb{R}}^{>0}$ such that for all non-torsion $u\in\Gamma$ we have
$\tau(u)>C$. Then $\gamma$ cannot be represented as $n$-th power for any
$n>C^{-1}\tau(\gamma)$. Let $m$ be a maximal number such that $\gamma$ is an
$m$-th power of an element $\gamma_{1}\in\Gamma$. Then $\gamma_{1}$ is
primitive.111We are grateful to Yves Cornulier and Sam Nead
(https://mathoverflow.net/users/1650/sam-nead) who suggested a proof of this
statement on Mathoverflow, [MO2].

###  4.2 HBG-quasimorphism associated with a connection

Definition 4.3:  A quasimorphism $q:\;\Gamma{\>\longrightarrow\>}G$ is called
homogeneous if $q$ restricted to any cyclic subgroup is a group homomorphism.

Let $M$ be a compact Riemannian manifold of strictly negative sectional
curvature, $\Gamma:=\pi_{1}(M)$, $\widetilde{M}$ its universal cover, and
$(B,\nabla)$ a bundle with connection.

By 4.1, every non-torsion element of $\pi_{1}(M)$ is a power of a primitive
element. Given a primitive $\gamma\in\pi_{1}(M)$, let $F_{\gamma}$ be the
shortest free geodesic loop representing $\gamma$. By [Kl, Theorem 3.8.14],
$F_{\gamma}$ is unique in every conjugacy class. For each conjugacy class of
$\gamma$ we fix a choice of a point $x\in F_{\gamma}$. When
$\gamma=\gamma_{1}^{d}$ and $\gamma_{1}$ is primitive, we denote by
$F_{\gamma}$ the loop $F_{\gamma_{1}}$ iterated $d$ times.

Fix a point $p\in M$, and $\gamma\in\pi_{1}(M)$. We are going to define a
homogeneous quasimorphism $q:\;\Gamma{\>\longrightarrow\>}GL(B_{p})$, where
$B_{p}$ denotes the fiber of $B$ in $p$, as follows.

Consider a point $x\in F_{\gamma}$, and let
$\widetilde{F}_{\gamma}:=\nu_{x,\gamma}\circ
F_{\gamma}\circ\nu^{-1}_{x,\gamma}$ be the 3-segment piecewise geodesic path
obtained by connecting $p$ to $x$, going around the loop $F_{\gamma}$ starting
and ending in $x$, and going back to $p$ along $\nu_{x,\gamma}$ in the
opposite direction. Clearly, this path represents $\gamma$ in $\pi_{1}(M,p)$.
Denote by $q(\gamma)\in GL(B_{p})$ the holonomy along
$\widetilde{F}_{\gamma}$. By construction, $q$ restricted to a cyclic subgroup
is always a homomorphism.

Note that $q$ depends on the choice of $x\in F_{\gamma}$, which has to be
fixed for each conjugacy class of $\gamma\in\Gamma$.

Theorem 4.4:  Let $M$ be a compact manifold with strictly negative sectional
curvature, $p\in M$ a base point, $\Gamma:=\pi_{1}(M)$, and
$q:\;\Gamma{\>\longrightarrow\>}GL(B_{p})$ the map defined above. Then
$q:\;\Gamma{\>\longrightarrow\>}GL(B_{p})$ is an Ulam quasimorphism. Moreover,
$q$ is homogeneous.

Proof: Let $\alpha,\beta,\gamma=(\alpha\beta)^{-1}$ be elements of $\Gamma$.
Choose any points $a\in F_{\alpha}$, $b\in F_{\beta}$, $c\in F_{\gamma}$. Then
$q(\alpha)q(\beta)q(\alpha\beta)^{-1}$ is a holonomy along a contractible
geodesic polygon with 9 edges obtained by going along

$\nu_{a,\alpha},F_{\alpha},\nu_{a,\alpha}^{-1},\nu_{b,\beta},F_{\beta},\nu_{b,\beta}^{-1},\nu_{c,\gamma},F_{\gamma},\nu_{c,\gamma}^{-1},$

see Figure 1. However, the holonomy along any contractible geodesic polygon is
bounded by 2.2. Homogeneity of $q$ is clear, because $q(\gamma^{n})$ is the
holonomy of $\nabla$ along the loop $\nu_{x,\gamma}\circ
F_{\gamma}^{n}\circ\nu_{x,\gamma}^{-1}$.

Figure 1:  Lift of the geodesic polygon to the universal cover.

Definition 4.5: Let $(M,\nabla)$ be a vector bundle with connection, and $q$
the Ulam quasimorphism defined in 4.2. Then
$q:\;\Gamma{\>\longrightarrow\>}GL(B_{p})$ is called the HBG-quasimorphism
associated with $(B,\nabla)$.

Remark 4.6: A homogeneous quasimorphism
$q:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}$ is unique in its equivalence
class ([C]). This is not known for quasimorphisms taking values in non-
commutative groups.

Remark 4.7: For any homogeneous quasimorphism
$q:\;\Gamma{\>\longrightarrow\>}G$ and $r\in G$, the map $rqr^{-1}$ is also a
homogeneous quasimorphism. The HBG-quasimorphism
$q:\;\pi_{1}(M,p){\>\longrightarrow\>}G$ associated with a $G$-bundle depends
on the choice of the points $x_{i}\in F_{\gamma_{i}}$ on the shortest loop
representing the primitive element $\gamma_{i}$. However, this dependence is
easy to describe: for any pair $x_{i},y_{i}\in F_{\gamma_{i}}$, consider the
geodesic triangle with vertices $x_{i},y_{i},p$, with the geodesic from
$x_{i}$ to $y_{i}$ going in the same direction as in the loop
$F_{\gamma_{i}}$. Let $r_{i}$ denote the loop along this triangle, starting in
$p$, going to $x$, then to $y$ and to $p$ again. Then $q(\gamma_{i})$ is the
holonomy along
$\nu_{x_{i},\gamma_{i}}F_{\gamma_{i}}\nu_{x_{i},\gamma_{i}}^{-1}$, or,
equivalently, the holonomy along
$r_{i}\nu_{y_{i},\gamma_{i}}F_{\gamma_{i}}\nu_{y_{i},\gamma_{i}}^{-1}r_{i}^{-1}$.
In other words, replacing the choice of $x_{i}$ by $y_{i}$ is equivalent to
the conjugation of $q(\gamma_{i})$ with the holonomy along a geodesic triangle
$r_{i}$, which is bounded. These two quasimorphisms are not always equivalent
(4.2), but they are algebraically equivalent (3.3).

In the commutative case, any homogeneous quasimorphism is conjugation
invariant ([C, §2.2.3]). The following result is a non-commutative Barge-Ghys
analogue of this statement.

Claim 4.8:  Let $q:\;\Gamma{\>\longrightarrow\>}G$ be a HBG-quasimorphism, and
$\alpha,\beta\in\Gamma$ conjugate elements. Then $q(\alpha)$ is conjugate to
$q(\beta)$.

Proof: Let $F_{\gamma}$ be the free geodesic loop homotopy equivalent to
$\alpha$ and $\beta$, and $x\in F_{\gamma}$ be the chosen point. Then
$q(\alpha)$, which is equal to holonomy of $\nabla$ along the loop
$\nu_{x,\alpha}\circ F_{\gamma}\circ\nu_{x,\alpha}^{-1}$, is conjugate to
$q(\beta)$, obtained the same way from the loop $\nu_{x,\beta}\circ
F_{\gamma}\circ\nu_{x,\beta}^{-1}$. It is easy to write this conjugation
explicitly: $q(\alpha)=Rq(\beta)R^{-1}$, where $R$ is holonomy of $\nabla$
along the loop $\nu_{x,\alpha}\circ\nu_{x,\beta}^{-1}$.

Question 4.9: Will 4.2 hold for an arbitrary homogeneous Ulam quasimorphism
$q:\;\Gamma{\>\longrightarrow\>}G$? When $G$ is commutative, this is true by
[C, §2.2.3].

Remark 4.10: For homogeneous quasimorphisms taking value in ${\mathbb{R}}$ or
${\mathbb{R}}^{n}$ (elsewhere, we call them “commutative”), equivalence
implies equality: if
$q_{1},q_{2}:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}^{n}$ are equivalent
homogeneous quasimorphisms, they satisfy $q_{1}=q_{2}$, because

$q_{1}(x)-q_{2}(x)=\lim_{n\to\infty}\frac{q_{1}(x^{n})-q_{2}(x^{n})}{n}=0.$

The same is true for all homogeneous Ulam quasimorphisms
$q:\;\Gamma{\>\longrightarrow\>}G$, if $G$ is nilpotent.

Claim 4.11:  Let $q_{1},q_{2}:\;\Gamma{\>\longrightarrow\>}G$ be homogeneous
quasimorphisms which are equivalent. Assume that $G$ is a simply connected
nilpotent Lie group. Then $q_{1}=q_{2}$.222We expect that this statement is
true whenever $G$ is a Lie group such that the exponential map
$\exp:\;{\mathfrak{g}}{\>\longrightarrow\>}G$ is a homeomorphism.

Proof. Step 1: Let $K\subset G$ be a compact subset such that
$q_{1}(x)q_{2}(x)^{-1}\in K$ for all $x\in\Gamma$. Since
$q_{i}(x^{n})=q_{i}(x)^{n}$, this gives $q_{1}(x)^{n}q_{2}(x)^{-n}\in K$. Then
4.2 would follow if we prove that for any distinct $a,b\in G$, the sequence
$a^{n}b^{-n}$ is unbounded, that is, does not belong to a compact set.

Step 2: Let $G_{i}$ be the lower central series for $G$. Applying induction
and using the long exact sequence

${\>\longrightarrow\>}\pi_{2}(G_{i}/G_{i-1}){\>\longrightarrow\>}\pi_{1}(G_{i-1}){\>\longrightarrow\>}\pi_{1}(G_{i}){\>\longrightarrow\>}\pi_{1}(G_{i}/G_{i-1}){\>\longrightarrow\>}0$

for the Serre’s fibration
$G_{i-1}{\>\longrightarrow\>}G_{i}{\>\longrightarrow\>}G_{i}/G_{i-1},$ we
immediately obtain that the groups $G_{i}$ and $G_{i}/G_{i-1}$ are simply
connected for all $i$. Since $G_{i}/G_{i-1}$ is abelian, this group is
isomorphic to ${\mathbb{R}}^{n}$.

Step 3: We use the induction on $\dim G$. When $G$ is 1-dimensional, this
group is commutative, giving $G={\mathbb{R}}$; if the sequence
$a^{n}b^{-n}\in{\mathbb{R}}$ is bounded, we have $a=b$.

Let $Z\subset G$ be the center; it is non-trivial, because $G$ is nilpotent.
Since $a^{n}b^{-n}$ is bounded modulo $Z$, we may apply the induction
hypothesis to the representatives of $a,b$ in $G/Z$ and obtain that $a=b$
modulo $Z$. Then $az=b$, where $z\in Z$, hence the sequence
$a^{n}b^{-n}=z^{n}$ is bounded. Since $Z={\mathbb{R}}^{n}$ (Step 2), this may
happen only if $z$ is trivial. Then $a=b$.

##  5 HBG-quasimorphisms with prescribed values

###  5.1 Connections with prescribed holonomy

We prove the following preliminary lemma, which will be used later in this
section.

Lemma 5.1:  Let $G$ be a connected Lie group, ${\mathfrak{g}}$ its Lie
algebra, and $P$ a trivial $G$-bundle on an interval $[0,1]$. Fix an element
$g\in G$. Denote by $\nabla_{0}$ the trivial connection on $P$. Then there
exists a ${\mathfrak{g}}$-valued 1-form $A$ with compact support, such that
the holonomy $\operatorname{Hol}(\nabla)$ of the connection
$\nabla:=\nabla_{0}+A$ is equal to $g$.

Proof: Write $A$ as $a(t)dt$, where $a\in{\mathfrak{g}}$ and $dt$ is the
standard 1-form on $[0,1]$. Then
$\operatorname{Hol}(\nabla)=\int_{0}^{1}a(t)dt$. Since $G$ is connected, we
can connect the unit $e$ to $g$ by a path
$\gamma:\;[0,1]{\>\longrightarrow\>}G$. Reparametrizing $\gamma$, we may
assume that $\gamma$ is constant in a small neighbourhood of $0$ and of $1$.
By Newton-Leibnitz formula,
$\int_{0}^{1}(\gamma(t)^{-1})^{*}\dot{\gamma}dt=g$. Setting
$a(t):=(\gamma(t)^{-1})^{*}\dot{\gamma}$, we obtain a connection form which
satisfies
$\operatorname{Hol}(\nabla)=\int_{0}^{1}(\gamma(t)^{-1})^{*}\dot{\gamma}dt=g$.
Since $\gamma$ is constant in a neighbourhood of $0$ and of $1$, the form
$a(t)dt$ has compact support.

###  5.2 Constructing the HBG-quasimorphisms

In this section, we prove 3.3. We repeat its statement for convenience.

3.3:  Let $M$ be a compact manifold of strictly negative curvature, $G$ a non-
abelian connected Lie group, and $x_{1},...,x_{n}\in\Gamma:=\pi_{1}(M)$ a
collection of primitive elements generating cyclic subgroups $U_{i}$
satisfying

$\forall u,v\in\Gamma,\forall i\neq j,\text{\ one has\ }U_{i}^{u}\cap
U_{j}^{v}=1$ (5.1)

where $U_{i}^{u}$ denotes the $u$-conjugate of $U_{i}$.111Since $\dim M>1$,
there exist infinitely many such subgroups, as follows from 3.2. Take any
collection of elements $g_{i}\in G$, with $i=1,...,n$. Then

(i)

There exists a connection $\nabla$ on a trivial principal bundle $P$ such that
the corresponding HBG-quasimorphism
$q_{\nabla}:\;\Gamma{\>\longrightarrow\>}G$ takes $x_{i}$ to $g_{i}$,
$i=1,2,...,n$.

(ii)

Moreover, for any countable family of cyclic subgroups $V_{i}\subset\Gamma$
satisfying

$\forall u,v\in\Gamma,\forall i\neq j,\text{\ one has\ }V_{i}^{u}\cap
U_{j}^{v}=1$

and $q_{\nabla}(V_{i})$ is not central in $G$.

Proof. Step 1: For each primitive element $x\in\pi_{1}(M)$, denote by $F_{x}$
the minimal free geodesic loop representing $x$ ([Kl, Theorem 3.8.14]). This
geodesic loop is unique and determines $x$ up to conjugation. Fix a point $p$
on $M$, and a point $p_{i}$ on each $F_{x_{i}}$. Let $\gamma_{x_{i}}$ be the
piecewise smooth loop connecting $p$ to $p_{i}$ by a geodesic segment
$\nu_{p,p_{i}}$, going around $F_{x_{i}}$, and back from $p_{i}$ to $p$ by the
same geodesic segment $\nu_{p,p_{i}}$ reversed. For each connection $\nabla$
on $P$, the corresponding HBG-quasimorphism takes $x_{i}$ to the holonomy of
$\nabla$ along $\gamma_{x_{i}}$.

We are going to choose a connection $\nabla$ on $P$ such that the monodromy of
$\nabla$ along $\gamma_{x_{i}}$ is equal to $g_{i}$.

Step 2: By (5.1), none of the loops $F_{x_{1}},...,F_{x_{n}}$ is contained in
another of these loops. Fix an open set $B_{x_{i}}$ containing a segment of
$F_{x_{i}}$ and not intersecting the rest of the loops. We can choose $B_{i}$
in such a way that it does not intersect the geodesic segment connecting $p$
to $p_{i}$ (Step 1). Denote by $\nabla_{0}$ the trivial connection on $P$.
Using 5.1, we modify $\nabla_{0}$ on each open set $B_{i}$ in such a way that
the monodromy of $\nabla$ along $\gamma_{i}\cap B_{i}$ is equal to $g_{i}$.

We built a connection $\nabla$ which satisfies 3.3 (i). To finish the proof of
3.3, it remains to show that $\nabla$ can be chosen in such a way that (ii) is
satisfied.

Step 3: Suppose that $\nabla$ is chosen in such a way that it satisfies 3.3
(i). Let $y_{m}$ denote one of the two generators of the cyclic subgroup
$V_{m}$. We modify the connection $\nabla$ on a sequence of small open sets
$D_{1},...,D_{k}$ with each $D_{l}$ not intersecting
$\gamma_{x_{1}},\gamma_{x_{2}},...,\gamma_{x_{n}}$, and
$\gamma_{y_{1}},\gamma_{y_{2}},...,\gamma_{y_{l-1}}$. The result of these
successive modifications is a connection denoted $\nabla_{k}$.

We choose $\nabla_{k}$ in such a way that $q_{\nabla_{k}}(V_{i})$ is not
central for $i=1,...,k$. The passage from $\nabla_{k-1}$ to $\nabla_{k}$ is
expressed by $\nabla_{k}=\nabla_{k-1}+\theta_{k}$, where $\theta_{k}$ is a
${\mathfrak{g}}$-valued 1-form with support in $D_{k}$.

We chose the 1-form $\theta_{k}$ very small, in such a way that the series
$\nabla+\sum\theta_{i}$ converges to a connection $\widetilde{\nabla}$. By
construction, the holonomy of $\widetilde{\nabla}$ along $\gamma_{y_{k}}$ is
equal to the holonomy of $\nabla_{k}$, hence $q_{\widetilde{\nabla}}(V_{k})$
is not central whenever $q_{\nabla_{k}}(V_{k})=q_{\widetilde{\nabla}}(V_{k})$
is not central.

The ${\mathfrak{g}}$-valued 1-form $\theta_{k}$ is chosen, on each step, using
the same argument as proves 5.1. Using this lemma, the 1-form $\theta_{k}$ is
determined by the value of $q_{\nabla_{k}}(V_{k})$, which can be chosen
arbitrarily small, provided that $q_{\nabla_{k}}(V_{k})$ is not central. Then
the series $\nabla+\sum_{i=1}^{\infty}\theta_{i}$ converges to a connection
$\widetilde{\nabla}$ which satisfies 3.3 (ii). Since the support of the form
$\sum_{i=1}^{\infty}\theta_{i}$ does not intersect the paths
$\gamma_{x_{1}},...,\gamma_{x_{n}}$, we have
$q_{\nabla}(x_{i})=q_{\widetilde{\nabla}}(x_{i})=g_{i}$, hence
$\widetilde{\nabla}$ satisfies both 3.3 (i) and 3.3 (ii).

###  5.3 $\varepsilon$-representations

As an application of the construction given in Subsection 5.2, we generalize
Kazhdan’s theorem [Kaz, Theorem 2] to the fundamental group of an arbitrary
compact manifold of negative sectional curvature. We define the notion of
$\varepsilon$-representations and $\delta$-approximated
$\varepsilon$-representation as follows (compare with [Kaz] and Subsection
1.5).

Let $G$ be a topological group. Recall that Chabauty topology on the set of
closed subgroups $C(G)$ is defined by the base of neighbourhoods,
$W_{U}(\Gamma)$ given for each open subset $U\subset G$ and a subgroup
$\Gamma\subset G$

$W_{U}(\Gamma)=\\{\Gamma^{\prime}\in C(G)\ \ |\ \
U\cdot\Gamma^{\prime}\supset\Gamma\text{\ \ and\ \
}U\cdot\Gamma\supset\Gamma^{\prime}\\}.$

If the topology of $G$ is induced by a left-invariant metric, the Chabauty
topology is induced by the Hausdorff metric on the space of subsets of $G$
(3.1).

By [F, p. 474], or [dH, Proposition 2 (vi)], the space $C(G)$ is compact.

Recall that a group $G$ has no small subgroups ([T]) if there exists a
neighbourhood of identity which does not contain a non-trivial subgroup; this
property is clearly satisfied by any Lie group. When $G$ has no small
subgroups, the space of $C_{0}(G)$ closed non-trivial subgroups is also
compact.

Clearly, the diameter of a subset of a metric space $M$ is a continuous
function in the topology on $2^{M}$ induced by the Hausdorff metric. Since
$C_{0}(G)$ is compact, the diameter of non-trivial subgroups in $G$ is bounded
from below for any metric on $G$.

Fix a left-invariant metric on $G$ such that any non-trivial subgroup has
diameter at least $1/3$. We motivate the choice of the constant $1/3$ as
follows. For compact group $G$, we can always choose a bi-invariant metric;
its geodesics are translations of one-parametric subgroups. We normalize the
metric such that the diameter of any compact one-parametric subgroup is
bounded from below by $1/2$.

Since every two points are connected by a geodesic, every element of a compact
group is an exponent of an element of the Lie algebra, and every finite cyclic
subgroup belongs to a circle subgroup. If we normalize the metric such that
the diameter of each circle subgroup is bounded from below by $1/2$, then the
diameter of each finite subgroup is at least $1/3$, which is realized by
${\mathbb{Z}}/3{\mathbb{Z}}$. In particular, of we choose the bi-invariant
metric on the group $SU(2)=S^{3}$ such that $\operatorname{\text{\sf
diam}}(SU(2))=1/2$ (this is equivalent to each meridian circle being of length
1), the bound $1/3$ is realized.

An $\varepsilon$-representation of a group $\Gamma$ is a map
$q:\;\Gamma{\>\longrightarrow\>}G$ such that $d(q(x)q(y),q(xy))<\varepsilon$
for any $x,y\in\Gamma$. An $\varepsilon$-representation can be
$\delta$-approximated by a representation if there exists a representation
$\rho:\;\Gamma{\>\longrightarrow\>}G$ such that $d(\rho(x),q(x))<\delta$ for
all $x\in\Gamma$.

Theorem 5.2:  Let $M$ be a compact manifold of negative sectional curvature,
$G$ a positive-dimensional connected Lie group, and $B$ a trivial principal
$G$-bundle. For any connection $\nabla$ in $B$, let
$q_{\nabla}:\;\pi_{1}(M){\>\longrightarrow\>}G$ denote the corresponding HBG-
quasimorphism (4.2). Choose a left-invariant metric on $G$ such that the
diameter of any closed subgroup is at least $1/3$. Then for each
$\varepsilon>0$, there exists a connection $\nabla$ such that $q_{\nabla}$ is
an $\varepsilon$-representation which cannot be $1/3$-approximated by a
representation.

Proof. Step 1: Let $a_{1},...,a_{n}$ be the generators of $\pi_{1}(M)$. Using
the same argument as in the proof of 3.3, we find $b\in\pi_{1}(M)$ such that
the cyclic group generated by $b$ does not intersect the cyclic groups
generated by $a_{i}$. Choose an open set $U_{b}$ which intersects $F_{b}$ and
does not intersect $F_{a_{1}},F_{a_{2}},...$ $\nu(p,p_{i})$, $i=1,...,n$,
where $\nu(p,p_{i})$ are geodesics connecting the marked point $p$ with
$F_{a_{i}}$. Let $p_{b}$ be a fixed point on $F_{b}$, and $\nabla_{0}$ be the
trivial connection on a trivial $G$-bundle $P$. Choose a connection
$\nabla_{0}+\theta$ which is trivial outside of $U_{b}$. Consider the path
$\gamma=\nu(p,p_{b})F_{b}\nu(p,p_{b})^{-1}$ as a map from $[0,1]$ to $M$. Then
the the holonomy of $\nabla+\theta$ along $\gamma$ is equal to
$\int_{0}^{1}\gamma^{*}(\theta)$. For any non-trivial $g=e^{u}$ in the image
of the exponential map, we can choose $\theta$ such that
$\gamma^{*}(\theta)=f(t)udt$, for some function
$f:\;[0,1]{\>\longrightarrow\>}{\mathbb{R}}$ with compact support. Choosing
$f$ such that $g=\int_{0}^{1}f(t)udt$, we obtain a 1-form $\theta$ with values
in the Lie algebra of $G$ such that the holonomy of $\nabla_{0}+\theta$ along
$\gamma$ is equal to $g$. Moreover, replacing $\theta$ with
$\frac{1}{m}\theta$, we obtain a connection with holonomy
$e^{\frac{1}{m}u}=g^{1/m}$.

For $m$ sufficiently large, this would give an $\varepsilon$-representation
$q_{\nabla}$ such that $q_{\nabla}(a_{i})=e$, and $q_{\nabla}(b)^{m}=g$.

Step 2: It remains to show that the $\varepsilon$-representation $q_{\nabla}$
cannot be $1/3$-aproximated by a representation $\rho$. By contradiction,
assume that $q_{\nabla}$ is $1/3$-aproximated by a representation
$\rho:\;\pi_{1}(M){\>\longrightarrow\>}G$. Since
$d(\rho(a_{i}^{n}),q_{\nabla}(a_{i})^{n})<1/3$, the closure of a subgroup of
$G$ generated by $\rho(a_{i})$ has diameter less than $1/3$. Since the
diameter of non-trivial subgroups of $G$ is $\geqslant 1/3$, this implies that
$\rho(a_{i})=\operatorname{Id}$, and $\rho$ is trivial. Therefore,
$\rho(b)=\operatorname{Id}$. However, for all $n\in{\mathbb{Z}}$, we have
$d(\rho(b)^{n},q_{\nabla}(b)^{n})<1/3$, because $q_{\nabla}$ is
$1/3$-approximated by $\rho$. Then the diameter of the subgroup of $G$
generated by $g=q_{\nabla}(b)$ is less than $1/3$, which again implies that
$g=\operatorname{Id}$, leading to contradiction.

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Michael Brandenbursky
Ben Gurion University of the Negev
Beer Sheva, Israel
<EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University HSE,
6 Usacheva Str. Moscow, Russia
<EMAIL_ADDRESS>