# Length functions on groups and rigidity

Shengkui Ye

###### Abstract

Let $G$ be a group. A function $l:G\rightarrow[0,\infty)$ is called a length
function if

(1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in\mathbb{Z};$

(2) $l(hgh^{-1})=l(g)$ for any $h,g\in G;$ and

(3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$

Such length functions exist in many branches of mathematics, mainly as stable
word lengths, stable norms, smooth measure-theoretic entropy, translation
lengths on $\mathrm{CAT}(0)$ spaces and Gromov $\delta$-hyperbolic spaces,
stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms,
dynamical degrees of birational maps and so on. We study length functions on
Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over
rings and Cremona groups. As applications, we prove that every group
homomorphism from an arithmetic subgroup of a simple algebraic
$\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index
subgroup of the elementary group $E_{n}(R)$ $(n\geq 3)$ over an associative
ring, or the Cremona group $\mathrm{Bir}(P_{\mathbb{C}}^{2})$ to any group $G$
having a purely positive length function must have its image finite. Here $G$
can be outer automorphism group $\mathrm{Out}(F_{n})$ of free groups, mapping
classes group $\mathrm{MCG}(\Sigma_{g})$, $\mathrm{CAT}(0)$ groups or Gromov
hyperbolic groups, or the group $\mathrm{Diff}(\Sigma,\omega)$ of
diffeomorphisms of a hyperbolic closed surface preserving an area form
$\omega.$

###  0.1 Introduction

The rigidity phenomena have been studied for many years. The famous Margulis
superrigidity implies any group homomorphism between irreducible lattices in
semisimple Lie groups of real rank $\mathrm{rk}_{\mathbb{R}}(G)\geq 2$ are
virtually induced by group homomorphisms between the Lie groups. Therefore,
group homomorphisms from ‘higher’-rank irreducible lattices to ‘lower’-rank
irreducible lattices normally have finite images. Farb, Kaimanovich and Masur
[26] [39] prove that every homomorphism from an (irreducible) higher rank
lattice into the mapping class group $\mathrm{MCG}(\Sigma_{g})$ has a finite
image. Bridson and Wade [17] showed that the same superrigidity remains true
if the target is replaced with the outer automorphism group
$\mathrm{Out}(F_{n})$ of the free group. Mimura [48] proves that every
homomorphism from Chevalley group over commutative rings to
$\mathrm{MCG}(\Sigma_{g})$ or $\mathrm{Out}(F_{n})$ has a finite image. Many
other rigidity results can be found, e.g. [49] [18] [19] [33] [54] and [53].
In this article, we study rigidity phenomena with the notion of length
functions.

Let $G$ be a group. We call a function $l:G\rightarrow[0,\infty)$ a length
function if

1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in\mathbb{Z};$

2) $l(aga^{-1})=l(g)$ for any $a,g\in G;$

3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$

Such length functions exist in geometric group theory, dynamical system,
algebra, algebraic geometry and many other branches of mathematics. For
example, the following functions $l$ are length functions (see Section 3 for
more examples with details).

  * •

(The stable word lengths) Let $G$ be a group generated by a symmetric (not
necessarily finite) set $S.$ For any $g\in G,$ the word length
$\phi_{S}(w)=\min\\{n\mid g=s_{1}s_{2}\cdots s_{n},$each $s_{i}\in S\\}$ is
the minimal number of elements of $S$ whose product is $g.$ The stable length
is defined as

$l(g)=\lim_{n\rightarrow\infty}\frac{\phi_{S}(g^{n})}{n}.$

  * •

(Stable norms) Let $M$ be a compact smooth manifold and $G=\mathrm{Diff}(M)$
the diffeomorphism group consisting of all self-diffeomorphisms. For any
diffeomorphism $f:M\rightarrow M,$ let

$\|f\|=\sup_{x\in M}\|D_{x}f\|,$

where $D_{x}f$ is the induced linear map between tangent spaces
$T_{x}M\rightarrow T_{f(x)}M.$ Define

$l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\|f^{n}\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\|f^{-n}\|}{n}\\}.$

  * •

(smooth measure-theoretic entropy) Let $M$ be a $C^{\infty}$ closed Riemannian
manifold and $G=\mathrm{Diff}_{\mu}^{2}(M)$ consisting of diffeomorphisms of
$M$ preserving a Borel probability measure $\mu.$ Let $l(f)=h_{\mu}(f)$ be the
measure-theoretic entropy, for any $f\in G=\mathrm{Diff}_{\mu}^{2}(M)$.

  * •

(Translation lengths) Let $(X,d)$ be a metric space and $G=\mathrm{Isom}(X)$
consisting of isometries $\gamma:X\rightarrow X$. Fix $x\in X,$ define

$l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$

This contains the translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov
$\delta$-hyperbolic spaces as special cases.

  * •

(average norm for quasi-cocycles) Let $G$ be a group and $E$ be a Hilbert
space with an $G$-action by linear isometrical action. A function
$f:G\rightarrow E$ is a quasi-cocyle if there exists $C>0$ such that

$\|f(gh)-f(g)-gf(h)\|<C$

for any $g,h\in G.$ Let $l:G\rightarrow[0,+\infty)$ be defined by

$l(g)=\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|}{n}.$

  * •

(Rotation numbers of circle homeomorphisms) Let $\mathbb{R}$ be the real line
and $G=\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})=\\{f\mid f:R\rightarrow R$ is a
monotonically increasing homeomorphism such that $f(x+n)=f(x)$ for any
$n\in\mathbb{Z}\\}.$ For any $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and
$x\in[0,1),$ the translation number is defined as

$l(f)=\lim_{n\rightarrow\infty}\frac{f^{n}(x)-x}{n}.$

  * •

(Asymptotic distortions) Let $f$ be a $C^{1+bv}$ diffeomorphism of the closed
interval $[0,1]$ or the circle $S^{1}.$ (“bv” means derivative with finite
total variation.) The asymptotic distortion of $f$ is defined (by Navas [50])
as

$l(f)=\lim_{n\rightarrow\infty}\mathrm{var}(\log Df^{n}).$

This gives a length function $l$ on the group $\mathrm{Diff}^{1+bv}(M)$ of
$C^{1+bv}$ diffeomorphisms for $M=[0,1]$ or $S^{1}.$

  * •

(Dynamical degree) Let $\mathbb{C}P^{n}$ be the complex projective space and
$f:\mathbb{C}P^{n}\dashrightarrow\mathbb{C}P^{n}$ be a birational map given by

$(x_{0}:x_{1}:\cdots:x_{n})\dashrightarrow(f_{0}:f_{1}:\cdots:f_{n}),$

where the $f_{i}$’s are homogeneous polynomials of the same degree without
common factors. The degree of $f$ is $\deg f=\deg f_{i}.$ Define

$l(f)=\max\\{\lim_{n\rightarrow\infty}\log\deg(f^{n})^{\frac{1}{n}},\lim_{n\rightarrow\infty}\log\deg(f^{-n})^{\frac{1}{n}}\\}.$

This gives a length function
$l:\mathrm{Bir}(\mathbb{C}P^{n})\rightarrow[0,+\infty).$ Here
$\mathrm{Bir}(\mathbb{C}P^{n})$ is the group of birational maps, also called
Cremona group.

The terminologies of length functions are used a lot in the literature (eg.
[28], [22]). However, they usually mean different things from ours (in
particular, it seems that the condition 3) has not been addressed for
commuting elements before).

Our first observation is the following result on vanishing of length
functions.

###### Theorem 0.1

Let $G_{A}=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ be an abelian-by-cyclic group,
where $A\in\mathrm{SL}_{2}(\mathbb{Z})$.

  1. (i)

When the absolute value of the trace $|\mathrm{tr}(A)|>2,$ any length function
$l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes
on $\mathbb{Z}^{2}.$

  2. (ii)

When $|\mathrm{tr}(A)|=2$ and $A\neq I_{2},$ any length function
$l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes
on the direct summand of $\mathbb{Z}^{2}$ spanned by eigenvectors of $A$.

###### Corollary 0.2

Suppose that the semi-direct product
$G_{A}=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ acts on a compact manifold by
Lipschitz homeomorphisms (or $C^{2}$-diffeomorphisms, resp.). The topological
entropy $h_{top}(g)=0$ (or Lyapunov exponents of $g$ are zero, resp.) for any
$g\in\mathbb{Z}^{2}$ when $|\mathrm{tr}(A)|>2$ or any eigenvector
$g\in\mathbb{Z}^{2}$ when $|\mathrm{tr}(A)|=2.$

It is well-known that the central element in the integral Heisenberg group
$G_{A}$ (for $A=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}$) is distorted in the
word metric. When the Heisenberg group $G_{A}$ acts on a $C^{\infty}$ compact
Riemannian manifold, Hu-Shi-Wang [36] proves that the topological entropy and
all Lyapunov exponents of the central element are zero. These results are
special cases of Theorem 0.1 and Corollary 0.2, by choosing special length
functions.

A length function $l:G\rightarrow[0,\infty)$ is called purely positive if
$l(g)>0$ for any infinite-order element $g.$ A group $G$ is called virtually
poly-positive, if there is a finite-index subgroup $H<G$ and a subnormal
series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=H$

such that every finitely generated subgroup of each quotient $H_{i}/H_{i+1}$
$(i=0,...,n-1)$ has a purely positive length function. Our following results
are on the rigidity of group homomorphisms.

###### Theorem 0.3

Let $\Gamma$ be an arithmetic subgroup of a simple algebraic
$\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2.$ Suppose that $G$ is
virtually poly-positive. Then any group homomorphism $f:\Gamma\rightarrow G$
has its image finite.

###### Theorem 0.4

Let $G$ be a group having a finite-index subgroup $H<G$ and a subnormal series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=H$

satisfying that

  1. (i)

every finitely generated subgroup of each quotient $H_{i}/H_{i+1}$
$(i=0,...,n-1)$ has a purely positive length function, i.e. $G$ is virtually
poly-positive; and

  2. (ii)

any torsion abelian subgroup in every finitely generated subgroup of each
quotient $H_{i}/H_{i+1}$ $(i=0,...,n-1)$ is finitely generated.

Let $R$ be a finitely generated associative ring with identity and $E_{n}(R)$
the elementary subgroup. Suppose that $\Gamma<E_{n}(R)$ is finite-index
subgroup. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image
finite when $n\geq 3$.

###### Corollary 0.5

Let $\Gamma$ be an arithmetic subgroup of a simple algebraic
$\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index
subgroup of the elementary subgroup $E_{n}(R)$ $(n\geq 3)$ for an associative
ring $R.$ Then any group homomorphism $f:E\rightarrow G$ has its image finite.
Here $G$ is one of the following groups:

  * •

a Gromov hyperbolic group,

  * •

$\mathrm{CAT(0)}$ group,

  * •

automorphism group $\mathrm{Aut}(F_{k})$ of a free group,

  * •

outer automorphism group $\mathrm{Out}(F_{k})$ of a free group or

  * •

mapping class group $\mathrm{MCG}(\Sigma_{g})$ $(g\geq 2)$.

  * •

the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a closed
surface preserving an area form $\omega.$

###### Theorem 0.6

Suppose that $G$ is virtually poly-positive. Let $R$ be a finitely generated
associative ring of characteristic zero such that any nonzero ideal is of a
finite index (eg. the ring of algebraic integers in a number field). Suppose
that $S<E_{n}(R)$ is a finite-index subgroup of the elementary group. Then any
group homomorphism $f:S\rightarrow G$ has its image finite when $n\geq 3$.

###### Corollary 0.7

Let $R$ be an associative ring of characteristic zero such that any nonzero
ideal is of a finite index. Any group homomorphism $f:E\rightarrow G$ has its
image finite, where $E<E_{n}(R)$ is finite-index subgroup and $n\geq 3$. Here
$G$ is one of the followings:

  * •

$\mathrm{Aut}(F_{k}),\mathrm{Out}(F_{k}),\mathrm{MCG}(\Sigma_{g}),$

  * •

a hyperbolic group,

  * •

a $\mathrm{CAT}(0)$ group or more generally a semi-hyperbolic group,

  * •

a group acting properly semi-simply on a $\mathrm{CAT}(0)$ space, or

  * •

a group acting properly semi-simply on a $\delta$-hyperbolic space,

  * •

the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a hyperbolic
closed surface preserving an area form $\omega.$

Some relevent cases of Theorem 0.4 and Theorem 0.6 are already established in
the literature. Bridson and Wade [17] showed that any group homomorphism from
an irreducible lattice in a semisimple Lie group of real rank $\geq 2$ to the
mapping class group $\mathrm{MCG}(\Sigma_{g})$ must have its image finite.
However, Theorem 0.3 can never hold when $\Gamma$ is a cocompact lattice,
since a cocompact lattice has its stable word length purely positive. When the
length functions involved in the virtually poly-positive group $G$ are
required to be stable word lengths, Theorem 0.3 holds more generally for
$\Gamma$ non-uniform irreducible lattices a semisimple Lie group of real rank
$\geq 2$ (see Proposition 8.4). When the length functions involved in the
virtually poly-positive group $G$ are given by a particular kind of quasi-
cocyles, Theorem 0.3 holds more generally for $\Gamma$ with property TT (cf.
Py [54], Prop. 2.2). Haettel [33] prove that any action of a high-rank a
higher rank lattice on a Gromov-hyperbolic space is elementary (i.e. either
elliptic or parabolic). Guirardel and Horbez [33] prove that every group
homomorphism from a high-rank lattice to the outer automorphism group of
torsion-free hyperbolic group has finite image. Thom [57] (Corollary 4.5)
proves that any group homomorphism from a boundedly generated with property
(T) to a Gromov hyperbolic group has finite image. Compared with these
results, our target group $G$ and the source group $E_{n}(R)$ (can be defined
over any non-commutative ring) in Theorem 0.4 are much more general. The
inequalities of $n$ in Theorem 0.4, Theorem 0.6 and Corollary 0.5, Corollary
0.7 can not be improved, since $\mathrm{SL}_{2}(\mathbb{Z})$ is hyperbolic.
The group $\Gamma$ in Corrolary 0.5 has Kazhdan’s property T (i.e. an
arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of
$\mathbb{Q}$-rank at least $2,$ or a finite-index subgroup of the elementary
subgroup $E_{n}(R),$ $n\geq 3,$ for an associative ring $R$ has Kazhdan’s
property T by [23]). However, there exist hyperbolic groups with Kazhdan’s
property $T$ (cf. [32], Section 5.6). This implies that Corrolary 0.5 does not
hold generally for groups $\Gamma$ with Kazhdan’s property T. Franks and
Handel [27] prove that any group homomorphism from a quasi-simple group
containing a subgroup isomorphic to the three-dimensional integer Heisenberg
group, to the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a
closed surface preserving an area form $\omega,$ has its image finite (cf.
Lemma 8.3).

We now study length functions on the Cremona groups.

###### Theorem 0.8

Let $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ $(n\geq 2)$ be the group of birational
maps on the projective space $\mathbb{P}_{k}^{n}$ over an algebraic closed
field $k$. Any length function
$l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,+\infty)$ vanishes on the
automorphism group $\mathrm{Aut}(\mathbb{P}_{k}^{n})=\mathrm{PGL}_{n+1}(k).$

When $n=2,$ a result of Blanc and Furter [9] (page 7 and Proposition 4.8.10)
implies that there are three length functions $l_{1},l_{2},l_{3}$ on
$\mathrm{Bir}(\mathbb{P}_{k}^{2})$ such that any element
$g\in\mathrm{Bir}(\mathbb{P}_{k}^{n})$ satisfying
$l_{1}(g)=l_{2}(g)=l_{3}(g)=0$ is either finite or conjugate to an element in
$\mathrm{Aut}(\mathbb{P}_{k}^{2}).$ This implies that the automorphism group
$\mathrm{Aut}(\mathbb{P}_{k}^{n})$ (when $k=2$) is one of the ‘largest’
subgroups of $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ on which every length function
vanishes.

###### Corollary 0.9

Let $G$ be a virtually poly-positive group. Any group homomorphism
$f:\mathrm{Bir}(\mathbb{P}_{k}^{2})\rightarrow G$ is trivial, for an algebraic
closed field $k$.

In particular, Corrolary 0.9 implies that any quotient group of
$\mathrm{Bir}(\mathbb{P}_{k}^{2})$ cannot act properly semisimply neither on a
Gromov $\delta$-hyperbolic space nor a $\mathrm{CAT}(0)$ space. This is
interesting, considering the following facts. There are (infinite-dimensional)
hyperbolic space and cubical complexes, on which $\mathrm{Bir}(P_{k}^{2})$
acts isometrically (see [21], Section 3.1.2 and [45]). The Cremona group
$\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is sub-quotient universal: every countable
group can be embedded in a quotient group of
$\mathrm{Bir}(\mathbb{P}_{k}^{2})$ (see [21], Theorem 4.7). Moreover, Blanc-
Lamy-Zimmermann [10] (Theorem E) proves that when $n\geq 3,$ there is a
surjection from $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ onto a free product of two-
element groups $\mathbb{Z}/2.$ This means that Corrolary 0.9 can never hold
for higher dimensional Cremona groups.

As byproducts, we give characterizations of length functions on Lie groups.
Our next result is that there is essentially only one length function on the
special linear group $\mathrm{SL}_{2}(\mathbb{R})$:

###### Theorem 0.10

Let $G=\mathrm{SL}_{2}(\mathbb{R}).$ Any length function
$l:G\rightarrow[0,+\infty)$ continuous on the subgroup $SO(2)$ and the
diagonal subgroup is proposional to the translation function

$\tau(g):=\inf_{x\in X}d(x,gx),$

where $X=\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SO}(2)$ is the upper-half plane.

More generally, we study length functions on Lie groups. Let $G$ be a
connected semisimple Lie group whose center is finite with an Iwasawa
decomposition $G=KAN$. Let $W$ be the Weyl group, i.e. the quotient group of
the normalizers $N_{K}(A)$ modulo the centralizers $C_{K}(A)$. Our second
result shows that a length function $l$ on $G$ is uniquely determined by its
image on $A.$

###### Theorem 0.11

Let $G$ be a connected semisimple Lie group whose center is finite with an
Iwasawa decomposition $G=KAN$. Let $W$ be the Weyl group.

  1. (i)

Any length function $l$ on $G$ that is continuous on the maximal compact
subgroup $K$ is determined by its image on $A.$

  2. (ii)

Conversely, any length function $l$ on $A$ that is $W$-invariant (i.e.
$l(w\cdot a)=l(a)$) can be extended to be a length function on $G$ that
vanishes on the maximal compact subgroup $K.$

The proofs of Theorems 0.10 0.11 are based the Jordan-Chevalley decompositions
of algebraic groups and Lie groups. We will prove that any length function on
a Heisenberg group vanishes on the central elements (see Lemma 5.2). This is a
key step for many other proofs. Based this fact, we prove Theorems 0.3, 0.6,
0.4, 0.8 by looking for Heisenberg subgroups. In Section 1, we give some
elementary facts on the length functions. In Section 2, we discuss typical
examples of length functions. In later sections, we study length functions on
Lie groups, algebraic groups, hyperbolic groups, matrix groups and the Cremona
groups.

##  1 Basic properties of length functions

###  1.1 Length functions

###### Definition 1.1

Let $G$ be a group. A function $l:G\rightarrow[0,\infty)$ is called a length
function if

1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in\mathbb{Z}.$

2) $l(aga^{-1})=l(g)$ for any $a,g\in G.$

3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$

###### Lemma 1.2

Any torsion element $g\in G$ has length $l(g)=0.$

Proof. Note that $l(1)=2l(1)$ and thus $l(1)=0.$ If $g^{n}=1,$ then
$l(g)=|n|l(1)=0.$

Recall that a subset $V$ of a real vector space is a convex cone, if $av+bw\in
V$ for any $v,w\in V$ and any non-negative real numbers $a,b\geq 0.$

###### Lemma 1.3

The set $\mathrm{Func}(G)$ of all length functions on a group $G$ is a convex
cone.

Proof. It is obvious that for two functions $l_{1},l_{2}$ on $G,$ a non-
negative linear combination $al_{1}+bl_{2}$ is a new length function.

###### Lemma 1.4

Let $f:G\rightarrow H$ be a group homomorphism between two groups $G$ and $H.$
For any length function $l:H\rightarrow[0,\infty),$ the composite $l\circ f$
is a length function on $G.$

Proof. It is enough to note that a group homomorphism preserves powers of
elements, conjugacy classes and commutativity of elements.

###### Corollary 1.5

For a group $G,$ let $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$ be the
outer automorphism group. Then $\mathrm{Out}(G)$ acts on the set
$\mathrm{Func}(G)$ of all length functions by pre-compositions

$l\mapsto l\circ g,$

where $l\in\mathrm{Func}(G),$ $g\in\mathrm{Out}(G)$. This action preserves
scalar multiplications and linear combinations (with non-negative
coefficients).

Proof. For an inner automorphism $I_{g}:G\rightarrow G$ given by
$I_{g}(h)=ghg^{-1},$ the length function $l\circ I_{g}=l$ since $l$ is
invariant under conjugation. Therefore, the outer automorphism group
$\mathrm{Out}(G)$ has an action on $\mathrm{Func}(G).$ It is obvious that the
pre-compositions preserve scalar multiplications and linear combinations with
non-negative coefficients.

###### Definition 1.6

A length function $l:G\rightarrow[0,\infty)$ is primitive if it is not a
composite $l^{\prime}\circ f$ for a non-trivial surjective group homomorphism
$f:G\twoheadrightarrow H$ and a length function $l^{\prime}:$
$H\rightarrow[0,\infty).$

###### Lemma 1.7

Suppose that a length function $l:G\rightarrow[0,\infty)$ vanishes on a
central subgroup $H<G.$ Then $l$ factors through the quotient group $G/H.$ In
other words, there exists a length function
$l^{\prime}:G/H\rightarrow[0,\infty)$ such that $l=l^{\prime}\circ q,$ where
$q:G\rightarrow G/H$ is the quotient group homomorphism.

Proof. Write $G=\cup gH,$ the union of left cosets. For any $h\in H,$ we have
$l(gh)\leq l(g)+l(h)=l(g)$ and $l(g)=l(ghh^{-1})\leq l(gh).$ Therefore,
$l(gh)=l(g)$ for any $h\in H.$ Define $l^{\prime}(gH)=l(g).$ Then $l^{\prime}$
is a length function on the quotient group $G/H.$ The required property
follows the definition easily.

###### Corollary 1.8

Suppose that a group $G$ has non-trivial finite central subgroup $Z(G).$ Any
length function $l$ on $G$ factors through $G/Z(G).$

Proof. This follows Lemma 1.7 and Lemma 1.2.

###### Lemma 1.9

Let $G$ be a group. Suppose that any non-trivial normal subgroup
$H\vartriangleleft G$ is of finite index. Then any non-vanishing length
function $l:G\rightarrow[0,\infty)$ is primitive.

Proof. Suppose that $l$ is a composite $l^{\prime}\circ f$ for a non-trivial
surjective group homomorphism $f:G\twoheadrightarrow H$ and a length function
$l^{\prime}:H\rightarrow[0,\infty).$ By the assumption of $G,$ the quotient
group $H$ is finite. This implies that $l^{\prime}$ and thus $l$ vanishes,
which is a contradiction.

###### Theorem 1.10

Let $\Gamma$ be an irreducible lattice in a connected irreducible semisimple
Lie group of real rank $\geq 2.$ Then any non-vanishing length function
$l:\Gamma\rightarrow[0,\infty)$ factors through a primitive function on
$\Gamma/Z(\Gamma)$.

Proof. By the Margulis-Kazhdan theorem (see [62], Theorem 8.1.2), any normal
subgroup $N$ of $\Gamma$ either lies in the center of $\Gamma$ (and hence it
is finite) or the quotient group $\Gamma/N$ is finite. Corollary 1.8 implies
that $l$ factors through a length function $l^{\prime}$ on $\Gamma/Z(\Gamma).$
The previous lemma 1.9 implies that $l^{\prime}$ is primitive.

##  2 Examples of length functions

Let’s see a general example first. Let $G$ be a goup and
$f:G\rightarrow[0,+\infty)$ be a function satisfying $f(gh)\leq f(g)+f(h)$ and
$f(g)=f(g^{-1})$ for any elements $g,h\in G.$ Define
$l:G\rightarrow[0,+\infty)$ by

$l(g)=\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}$

for any $g\in G.$

###### Lemma 2.1

The function $l$ is a length function in the sense of Definition 1.1.

Proof. For any $g\in G,$and natural numbers $n,m,$ we have $f(g^{n+m})\leq
f(g^{n})+f(g^{m}).$ This means that $\\{f(g^{n})\\}_{n=1}^{\infty}$ is a
subadditive sequence and thus the limit
$\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}$ exists. This shows that $l$ is
well-defined.

From the definition of $l,$ it is clear that $l(g^{n})=|n|l(g)$ for any
integer $n.$ Let $h\in G.$ We have

$l(hgh^{-1})=\lim_{n\rightarrow\infty}\frac{f(hg^{n}h^{-1})}{n}\leq\lim_{n\rightarrow\infty}\frac{f(h)+f(g^{n})+f(h^{-1})}{n}=\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}=l(g).$

Similarly, we have $l(g)=l(h^{-1}(hgh^{-1})h)\leq l(hgh^{-1})$ and thus
$l(g)=l(hgh^{-1}).$ For commuting elements $a,b,$ we have
$(ab)^{n}=a^{n}b^{n}.$ Therefore,

$\displaystyle l(ab)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{f((ab)^{n})}{n}=\lim_{n\rightarrow\infty}\frac{f(a^{n}b^{n})}{n}$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\frac{f(a^{n})+f(b^{n})}{n}\leq
l(a)+l(b).$

Many (but not all) length functions $l$ come from subadditive functions $f.$

###  2.1 Stable word lengths

Let $G$ be a group generated by a (not necessarily finite) set $S$ satisfying
$s^{-1}\in S$ for each $s\in S.$ For any $g\in G,$ the word length
$\phi_{S}(w)=\min\\{n\mid g=s_{1}s_{2}\cdots s_{n},$each $s_{i}\in S\\}$ is
the minimal number of elements of $S$ whose product is $g.$ The stable length
$l(g)=\lim_{n\rightarrow\infty}\frac{\phi_{S}(g^{n})}{n}.$ Since
$\phi_{S}(g^{n})$ is subadditive, the limit always exists.

###### Lemma 2.2

The stable length $l:G\rightarrow[0,+\infty)$ is a length function in the
sense of Definition 1.1.

Proof. From the definition of the word length $\phi_{S},$ it is clear that
$\phi_{S}(gh)\leq\phi_{S}(g)+\phi_{S}(h)$ and $\phi_{S}(g)=\phi_{S}(g^{-1})$
for any $g,h\in G.$ The claim is proved by Lemma 2.1.

When $S$ is the set of commutators, the $l(g)$ is called the stable commutator
length, which is related to lots of topics in low-dimensional topology (see
Calegari [20]).

###  2.2 Growth rate

Let $G$ be a group generated by a finite set $S$ satisfying $s^{-1}\in S$ for
each $s\in S.$ Suppose $|\cdot|_{S}$ is the word length of $(G,S).$ For any
automorphism $\alpha:G\rightarrow G,$ define
$l^{\prime}(\alpha)=\max\\{|\alpha(s_{i})|_{S}:s_{i}\in S\\}.$ Let
$l(\alpha)=\lim_{n\rightarrow\infty}\frac{\log l^{\prime}(\alpha^{n})}{n}.$
This number $l(\alpha)$ is called the algebraic entropy of $\alpha$ (cf. [40],
Definition 3.1.9, page 114).

###### Lemma 2.3

Let $\mathrm{Aut}(G)$ be the group of automorphisms of $G.$ The function
$l:\mathrm{Aut}(G)\rightarrow[0,+\infty)$ is a length function in the sense of
Definition 1.1.

Proof. Since $\alpha(s_{i})^{-1}=\alpha^{-1}(s_{i})$ for any $s_{i}\in S,$ we
know that $l^{\prime}(\alpha)=l^{\prime}(\alpha^{-1}).$ For another
automorphism $\beta:G\rightarrow G,$ let
$l^{\prime}(\beta)=|\beta(s_{i})|_{S}$ for some $s_{i}\in S.$ Suppose that
$\beta(s_{i})=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ with $k=l^{\prime}(\beta).$
Then
$|(\alpha\beta)(s_{i})|_{S}=|\alpha(s_{i_{1}})\alpha(s_{i_{2}})\cdots\alpha(s_{i_{k}})|_{S}\leq
l^{\prime}(\alpha)k.$ This proves that $l^{\prime}(\alpha\beta)\leq
l^{\prime}(\alpha)l^{\prime}(\beta).$ The claim is proved by Lemma 2.1.

Fix $g\in G.$ For any automorphism $\alpha:G\rightarrow G,$ define
$b_{n}=|\alpha^{n}(g)|_{S}.$ Suppose that $g=s_{1}s_{2}\cdots s_{k}$ with
$k=|g|_{S}.$ Note that
$b_{n}=|\alpha^{n}(g)|_{S}=|\alpha^{n}(s_{1})\alpha^{n}(s_{2})\cdots\alpha^{n}(s_{k})|_{S}\leq
l^{\prime}(\alpha^{n})|g|_{S}.$ Therefore, we have

$\lim\sup_{n\rightarrow\infty}\frac{\log b_{n}}{n}\leq l(\alpha).$

This implies that $l(\alpha)$ is an upper bounded for growth rate of
$\\{|\alpha^{n}(g)|_{S}\\}.$ The growth rate is studied a lot in geometric
group theory (for example, see [43] for growth of automorphisms of free
groups).

###  2.3 Matrix norms and group acting on smooth manifolds

For a square matrix $A,$ the matrix norm $\|A\|=\sup_{\|x\|=1}\|Ax\|.$ Define
the stable norm $s(A)=\lim_{n\rightarrow+\infty}\frac{\log\|A^{n}\|}{n}.$
Since $\|AB\|\leq\|A\|\|B\|$ for any two matrices $A,B,$ the sequence
$\\{\log\|A^{n}\|\\}_{n=1}^{\infty}$ is subadditive and thus the limit exists.

###### Lemma 2.4

Let $G=\mathrm{GL}_{n}(\mathbb{R})$ be the general linear group. The function
$l:G\rightarrow[0,+\infty)$ defined by

$l(g)=\max\\{s(g),s(g^{-1})\\}$

is a length function in the sense of Definition 1.1.

Proof. From the definition of the matrix norm, it is clear that
$\log\|gh\|\leq\log\|g\|+\log\|h\|$ for any $g,h\in G.$ Then
$l(g)=\max\\{s(g),s(g^{-1})\\}$ is a length function by Lemma 2.1.

Let $M$ be a compact smooth manifold and $\mathrm{Diff}(M)$ the diffeomorphism
group consisting of all self-diffeomorphisms. For any diffeomorphism
$f:M\rightarrow M,$ let

$\|f\|=\sup_{x\in M}\|D_{x}f\|,$

where $D_{x}f$ is the induced linear map between tangent spaces
$T_{x}M\rightarrow T_{f(x)}M.$ Define

$l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\|f^{n}\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\|f^{-n}\|}{n}\\}.$

A similar argument as the proof of the previous lemma proves the following.

###### Lemma 2.5

Let $G$ be a group acting on a smooth manifold $M$ by diffeomorphisms. The
function $l:G\rightarrow[0,+\infty)$ is a length function in the sense of
Definition 1.1.

For an $f$-invariant Borel probability measure $\mu$ on $M,$ it is well known
(see [O]) that there exists a measurable subset $\Gamma_{f}\subset M$ with
$\mu(\Gamma_{f})=1$ such that for all $x\in\Gamma_{f}$ and $u\in T_{x}M,$ the
limit

$\chi(x,u,f)=\lim\frac{1}{n}\log\|D_{x}f^{n}(u)\|$

exists and is called Lyapunov exponent of $u$ at $x.$ From the definitions, we
know that $\chi(x,u,f)\leq l(f)$ for any $x\in\Gamma_{f}$ and $u\in T_{x}M.$

###  2.4 Smooth measure-theoretic entropy

Let $T:X\rightarrow X$ be a measure-preserving map of the probability space
$(X,\mathfrak{B},m).$ For a finite-sub-$\sigma$-algebra
$A=\\{A_{1},A_{2},...,A_{k}\\}$ of $\mathfrak{B},$ denote by

$\displaystyle H(A)$ $\displaystyle=$ $\displaystyle-\sum m(A_{i})\log
m(A_{i}),$ $\displaystyle h(T,A)$ $\displaystyle=$
$\displaystyle\lim\frac{1}{n}H(\vee_{i=0}^{n-1}T^{-i}A),$

where $\vee_{i=0}^{n-1}T^{-i}A$ is a set consisting of sets of the form
$\cap_{i=0}^{n-1}T^{-i}A_{j_{i}}.$ The entropy of $T$ is defined as
$h_{m}(T)=\sup h(T,A),$ where the supremum is taken over all finite sub-
algebra $A$ of $\mathfrak{B}.$ For more details, see Walters [58] (Section
4.4).

###### Lemma 2.6

Let $M$ be a $C^{\infty}$ closed Riemannian manifold and
$G=\mathrm{Diff}_{\mu}^{2}(M)$ consisting of diffeomorphisms of $M$ preserving
a Borel probability measure $\mu.$ The entropy $h_{\mu}$ is a length function
on $\mathrm{Diff}_{\mu}^{2}(M)$ in the sense of Definition 1.1.

Proof. For any $f,g\in\mathrm{Diff}_{\mu}^{2}(M)$ and integer $n,$ it is well-
known that $h_{\mu}(f^{n})=|n|h_{\mu}(f)$ and $h_{\mu}(f)=h_{\mu}(gfg^{-1})$
(cf. [58], Theorem 4.11 and Theorem 4.13). Hu [35] proves that
$h_{\mu}(fg)\leq h_{\mu}(f)+h_{\mu}(g)$ when $fg=gf.$

###  2.5 Stable translation length on metric spaces

Let $(X,d)$ be a metric space and $\gamma:X\rightarrow X$ an isometry. Fix
$x\in X.$ Note that $d(x,\gamma_{1}\gamma_{2}x)\leq
d(x,\gamma_{1}x)+d(\gamma_{1}x,\gamma_{1}\gamma_{2}x)=d(x,\gamma_{1}x)+d(x,\gamma_{2}x)$
and $d(x,\gamma_{1}x)=d(x,\gamma_{1}^{-1}x)$ for any isometries
$\gamma_{1},\gamma_{2}.$ Define

$l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$

For any $y\in X,$ we have

$\displaystyle d(x,\gamma^{n}x)$ $\displaystyle\leq$ $\displaystyle
d(x,y)+d(y,\gamma^{n}y)+d(\gamma^{n}y,\gamma^{n}x)$ $\displaystyle=$
$\displaystyle 2d(x,y)+d(y,\gamma^{n}y)$

and thus
$\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}\leq\lim_{n\rightarrow\infty}\frac{d(y,\gamma^{n}y)}{n}.$
Similarly, we have the other direction

$\lim_{n\rightarrow\infty}\frac{d(y,\gamma^{n}y)}{n}\leq\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$

This shows that the definition of $l(\gamma)$ does not depend on the choice of
$x.$

###### Lemma 2.7

Let $G$ be a group acting isometrically on a metric space $X.$ Then the
function $l:G\rightarrow[0,+\infty)$ defined by $g\longmapsto l(g)$ as above
is a length function in the sense of Definition 1.1.

Proof. This follows Lemma 2.1.

###  2.6 Translation lengths of isometries of CAT(0) spaces

In this subsection, we will prove that the translation length on a CAT(0)
space defines a length function. First, let us introduce some notations. Let
$(X,d_{X})$ be a geodesic metric space, i.e. any two points $x,y\in X$ can be
connected by a path $[x,y]$ of length $d_{X}(x,y)$. For three points $x,y,z\in
X,$ the geodesic triangle $\Delta(x,y,z)$ consists of the three vertices
$x,y,z$ and the three geodesics $[x,y],[y,z]$ and $[z,x].$ Let
$\mathbb{R}^{2}$ be the Euclidean plane with the standard distance
$d_{\mathbb{R}^{2}}$ and $\bar{\Delta}$ a triangle in $\mathbb{R}^{2}$ with
the same edge lengths as $\Delta$. Denote by
$\varphi:\Delta\rightarrow\bar{\Delta}$ the map sending each edge of $\Delta$
to the corresponding edge of $\bar{\Delta}.$ The space $X$ is called a CAT(0)
space if for any triangle $\Delta$ and two elements $a,b\in\Delta,$ we have
the inequality

$d_{X}(a,b)\leq d_{\mathbb{R}^{2}}(\varphi(a),\varphi(b)).$

The typical examples of CAT(0) spaces include simplicial trees, hyperbolic
spaces, products of CAT(0) spaces and so on. From now on, we assume that $X$
is a complete CAT(0) space. Denote by Isom$(X)$ the isometry group of $X.$ For
any $g\in$ Isom$(X)$, let

$\mathrm{Minset}(g)=\\{x\in X:d(x,gx)\leq d(y,gy)\text{ for any }y\in X\\}$

and let $\tau(g)=\inf\nolimits_{x\in X}d(x,gx)$ be the translation length of
$g.$ When the fixed-point set $\mathrm{Fix}(g)\neq\emptyset,$ we call $g$
elliptic. When $\mathrm{Minset}(g)\neq\emptyset$ and $d_{X}(x,gx)=\tau(g)>0$
for any $x\in\mathrm{Minset}(g),$ we call $g$ hyperbolic. The group element
$g$ is called semisimple if the minimal set $\mathrm{Minset}(g)$ is not empty,
i.e. it is either elliptic or hyperbolic. A subset $C$ of a CAT(0) space if
convex, if any two points $x,y\in C$ can connected by the geodesic segment
$[x,y]\subset C.$ A group $G$ is called CAT(0) if $G$ acts properly
discontinuously and cocompactly on a CAT(0) space $X$. In such a case, any
infinite-order element in $G$ acts hyperbolically on $X.$ For more details on
CAT(0) spaces, see the book of Bridson and Haefliger [16].

The following was proved by Ballmann-Gromov-Schroeder [2] (Lemma 6.6, page
83). The original proof was for Hardmard manifolds, which also holds for
general cases. For completeness, we give details here.

###### Lemma 2.8

Let $\gamma:X\rightarrow X$ be an isometry of a complete CAT(0) space $X.$ For
any $x_{0}\in X,$ we have

$\tau(\gamma):=\inf_{x\in X}d(\gamma
x,x)=\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}x_{0},x_{0})}{k}.$

Proof. For any $p=x_{0}\in X,$ let $m$ be the middle point of $[p,\gamma p].$
We have that $d(m,\gamma m)\leq\frac{1}{2}d(p,\gamma^{2}p)$ by the convexity
of length functions. Therefore, $d(p,\gamma^{2}p)\geq 2\tau(\gamma)$ and
$\tau(\gamma^{2})\geq 2\tau(\gamma).$ Note that $d(p,\gamma^{2}p)\leq
d(p,\gamma p)+d(\gamma p,\gamma^{2}p)=2d(p,\gamma p)$ and thus
$\tau(\gamma^{2})\leq 2\tau(\gamma).$ Inductively, we have

$2^{n}\tau(\gamma)\leq d(p,\gamma^{2^{n}}p)\leq 2^{n}d(p,\gamma p).$

Note that the limit $\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}p,p)}{k}$
exists and is independent of $p$ (see the previous subsection). Therefore, the
limit $\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}p,p)}{k}$ equals to
$\tau(\gamma).$

###### Corollary 2.9

Let $X$ be a complete CAT(0) space and $G$ a group acting on $X$ by
isometries. For any $g\in G,$ define $\tau(g)=\inf_{x\in X}d(x,gx)$ as the
translation length. Then $\tau:G\rightarrow[0,+\infty)$ is a length function
in the sense of Definition 1.1.

Proof. This follows Lemma 2.8 and Lemma 2.7.

###  2.7 Translation lengths of Gromov $\delta$-hyperbolic spaces

Let $\delta>0.$ A geodesic metric space $X$ is called Gromov
$\delta$-hyperbolic if for any geodesic triangle $\Delta xyz$ one side $[x,y]$
is contained a $\delta$-neighborhood of the other two edges $[x,z]\cup[y,z].$
Fix $x_{0}\in X.$ Any isometry $\gamma:X\rightarrow X$ is called elliptic if
$\\{\gamma^{n}x\\}_{ne\mathbb{Z}}$ is bounded. If the orbit map
$\mathbb{Z}\rightarrow X$ given by $n\mapsto\gamma^{n}x_{0}$ is quasi-
isometric (i.e. there exists $A\geq 1$ and $B\geq 0$ such that

$\frac{1}{A}|n-m|-B\leq d_{X}(\gamma^{n}x_{0},\gamma^{m}x_{0})\leq A|n-m|+B$

for any integers $n,m$), we call $\gamma$ is hyperbolic. Otherwise, we call
$\gamma$ is parabolic. Define
$l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(\gamma^{n}x_{0},x_{0})}{n}.$ For
any group $G$ acts isometrically on a $\delta$-hyperbolic space, the function
$l:G\rightarrow[0,\infty)$ is a length function by Lemma 2.7. A finitely
generated group $G$ is Gromov $\delta$-hyperbolic if for some finite
generating set $S,$ the Caley graph $\Gamma(G,S)$ is Gromov
$\delta$-hyperbolic. Any infinite-order element $g$ in a Gromov
$\delta$-hyperbolic group is hyperbolic and thus has positive length $l(g)>0$
(cf. [32], 8.1.D). For more details on hyperbolic spaces and hyperbolic
groups, see the book [32] of Gromov.

###  2.8 Quasi-cocycles

Let $G$ be a group and $(E,\|\cdot\|)$ be a normed vector space with an
$G$-action by linear isometries. A function $f:G\rightarrow E$ is a quasi-
cocyle if there exists $C>0$ such that

$\|f(gh)-f(g)-gf(h)\|<C$

for any $g,h\in G.$ Let $l:G\rightarrow[0,+\infty)$ be defined by

$l(g)=\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|}{n}.$

Note that $\|f(g^{n+m})\|\leq\|f(g^{n})\|+\|f(g^{m})\|+C$ for any integers
$n,m\geq 0.$ This general subadditive property implies that the limit
$\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|}{n}$ exists (see [56], Theorem
1.9.2, page 22). We call $l$ the average norm. Many applications of quasi-
cocycles can be found in [49].

###### Lemma 2.10

For any quasi-cocyle $f:G\rightarrow E,$ the average norm $l$ is a length
function.

Proof. For any natural number $n,$ we have

$\|f(1)-f(g^{-n})-g^{-n}f(g^{n})\|<C$

and thus $\|\frac{f(1)-f(g^{-n})-g^{-n}f(g^{n})}{n}\|<\frac{C}{n}.$ Taking the
limit, we have
$\lim_{n\rightarrow\infty}\frac{\|f(g^{-n})\|}{n}=\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|}{n}.$
Therefore, for any $k\in\mathbb{Z},$ we have
$l(g^{k})=\lim_{n\rightarrow\infty}\frac{\|f(g^{kn})\|}{n}=|k|l(g).$ For any
$h\in G,$ we have

$\|f(hg^{n}h^{-1})\|\leq\|f(h)\|+\|f(h^{-1})\|+\|f(g^{n})\|+2C.$

Therefore, we have

$l(hgh^{-1})=\lim_{n\rightarrow\infty}\frac{\|f(hg^{n}h^{-1})\|}{n}\leq\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|}{n}=l(g).$

Similarly, we have $l(g)=l(h^{-1}(hgh^{-1})h)\leq l(hgh^{-1}).$ When $g,h$
commutes, we have

$\displaystyle l(gh)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{\|f((gh)^{n})\|}{n}=\lim_{n\rightarrow\infty}\frac{\|f(g^{n}h^{n})\|}{n}$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\frac{\|f(g^{n})\|+\|f(h^{n})\|+C}{n}=l(g)+l(h).$

###  2.9 Rotation number

Let $\mathbb{R}$ be the real line and
$\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})=\\{f\mid
f:\mathbb{R}\rightarrow\mathbb{R}$ is a monotonically increasing homeomorphism
such that $f(x+n)=f(x)$ for any $n\in\mathbb{Z}\\}.$ For any
$f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and $x\in[0,1),$ the translation
number is defined as

$l(f)=\lim_{n\rightarrow\infty}\frac{f^{n}(x)-x}{n}.$

It is well-known that $l(f)$ exists and is independent of $x$ (see [51], Prop.
2.22, p.31). Note that every $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$
induces an orientation-preserving homeomorphism of the circle $S^{1}.$

###### Proposition 2.11

The absolute value of the translation number
$|l|:\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})\rightarrow[0,\infty)$ is a length
function in the sense of Definition 1.1.

Proof. For any $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and
$k\in\mathbb{Z}\backslash\\{0\\},$ we have that

$l(f^{k})=\lim_{n\rightarrow\infty}\frac{f^{kn}(x)-x}{n}=k\lim_{n\rightarrow\infty}\frac{f^{kn}(x)-x}{nk}=kl(f).$

For any $a\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R}),$ we have that

$\displaystyle\mid$ $\displaystyle
l(afa^{-1})-l(f)\mid=\lim_{n\rightarrow\infty}\mid\frac{af^{n}(a^{-1}x)-x-f^{n}(x)+x}{n}\mid$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\mid\frac{af^{n}(a^{-1}x)-f^{n}(a^{-1}x)+f^{n}(a^{-1}x)-f^{n}(x)}{n}\mid$
$\displaystyle=$ $\displaystyle 0,$

since $a$ is bounded on $[0,1]$ and $\mid f^{n}(a^{-1}x)-f^{n}(x)\mid\leq
2+|a^{-1}x-x|.$

For commuting elements $f,g\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R}),$ we have
that

$l(fg)=\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}(x))-x}{n}=\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}(x))-g^{n}(x)+g^{n}(x)-x}{n}.$

Suppose that $g^{n}(x)=k_{n}+x_{n}$ for $k_{n}\in\mathbb{Z}$ and
$x_{n}\in[0,1).$ Then

$\displaystyle\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}x)-g^{n}(x)+g^{n}(x)-x}{n}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{f^{n}(0)-0+g^{n}(x)-x}{n}=l(f)+l(g).$

Therefore, we get $|l(fg)|\leq|l(f)|+|l(g)|.$

###### Remark 2.12

It is actually true that the rotation number $l$ is multiplicative on any
amenable group (see [51], Prop. 2.2.11 and the proof of Prop. 2.2.10, page
36). This implies that the absolute rotation number $|l|$ is subadditive on
any amenable group. In other words, for any amenable group
$G<\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and any $g,h\in G$ we have
$|l(gh)|\leq|l(g)|+|l(h)|.$

###  2.10 Asymptotic distortions

Let $f$ be a $C^{1+bv}$ diffeomorphism of the closed interval $[0,1]$ or the
circle $S^{1}.$ (“bv” means derivative with finite total variation.) The
asymptotic distortion of $f$ is defined as

$l(f)=\mathrm{dist}_{\infty}(f)=\lim_{n\rightarrow\infty}\mathrm{var}(\log
Df^{n}).$

It’s proved by Eynard-Bontemps amd Navas ([24], pages 7-8) that

  1. (1)

$\mathrm{dist}_{\infty}(f^{n})=|n|\mathrm{dist}_{\infty}(f)$ for all
$n\in\mathbb{Z}$;

  2. (2)

$\mathrm{dist}_{\infty}(hfh^{-1})=\mathrm{dist}_{\infty}(f)$ for every
$C^{1+bv}$ diffeomorphism $h;$

  3. (3)

$\mathrm{dist}_{\infty}(f\circ
g)\leq\mathrm{dist}_{\infty}(f)+\mathrm{dist}_{\infty}(g)$ for commuting
$f,g.$

Therefore, the asymptotic distortion is a length function $l$ on the group
$\mathrm{Diff}^{1+bv}(M)$ of $C^{1+bv}$ diffeomorphisms for $M=[0,1]$ or
$S^{1}.$

###  2.11 Dynamical degrees of Cremona groups

Let $k$ be a field and
$\mathbb{P}_{k}^{n}=k^{n+1}\backslash\\{0\\}/\\{\lambda\sim\lambda
x:\lambda\neq 0\\}$ be the projective space. A rational map from
$\mathbb{P}_{k}^{n}$ to itself is a map of the following type

$(x_{0}:x_{1}:\cdots:x_{n})\dashrightarrow(f_{0}:f_{1}:\cdots:f_{n})$

where the $f_{i}$’s are homogeneous polynomials of the same degree without
common factor. The degree of $f$ is $\deg f=\deg f_{i}.$ A birational map from
$\mathbb{P}_{k}^{n}$ to itself is a rational map
$f:\mathbb{P}_{k}^{n}\dashrightarrow\mathbb{P}_{k}^{n}$ such that there exists
a rational map $g:\mathbb{P}_{k}^{n}\dashrightarrow\mathbb{P}_{k}^{n}$ such
that $f\circ g=g\circ f=\mathrm{id}.$ The group
$\mathrm{Bir}(\mathbb{P}_{k}^{n})$ of birational maps is called the Cremona
group (also denoted as $\mathrm{Cr}_{n}(k)$). It is well-known that
$\mathrm{Bir}(\mathbb{P}_{k}^{n})$ is isomorphic to the group
$\mathrm{Aut}_{k}(k(x_{1},x_{2},\cdots,x_{n}))$ of self-isomorphisms of the
field $k(x_{1},x_{2},\cdots,x_{n})$ of the rational functions in $n$
indeterminates over $k.$ The (first) dynamical degree $\lambda(f)$ of
$f\in\mathrm{Bir}(\mathbb{P}_{k}^{n})$ is defined as

$\lambda(f)=\max\\{\lim_{n\rightarrow\infty}\deg(f^{n})^{\frac{1}{n}},\lim_{n\rightarrow\infty}\deg(f^{-n})^{\frac{1}{n}}\\}.$

Since $\deg(f^{n})^{\frac{1}{n}}$ is sub-multiplicative, the limit exists.

###### Lemma 2.13

Let $l(f)=\log\lambda(f).$ Then
$l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,+\infty)$ is a length
function.

Proof. Without loss of generality, we assume that
$\lambda(f)=\lim_{n\rightarrow\infty}\deg(f^{n})^{\frac{1}{n}},$ while the
other case can be considered similarly. For any $k\in\mathbb{N},$ it is easy
that $l(f^{k})=\lim_{n\rightarrow\infty}\frac{\log\deg f^{nk}}{n}=kl(f).$ For
any $h\in\mathrm{Bir}(\mathbb{P}_{k}^{n}),$ we have

$l(hfh^{-1})=\lim_{n\rightarrow\infty}\frac{\log\deg
hf^{n}h^{-1}}{n}=\lim_{n\rightarrow\infty}\frac{\log\deg f^{nk}}{n}=l(f).$

For commuting maps $f,g,$ we have $(fg)^{n}=f^{n}g^{n}.$ Therefore,

$\displaystyle l(fg)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{\log\deg f^{n}g^{n}}{n}$
$\displaystyle\leq$ $\displaystyle\lim_{n\rightarrow\infty}\frac{\log\deg
f^{n}}{n}+\lim_{n\rightarrow\infty}\frac{\log\deg g^{n}}{n}=l(f)+l(g).$

This checks the three conditions of the length function.

It is surprising that when $n=2$ and $k$ is an algebraically closed field, the
length function $l(f)$ is given by the translation length $\tau(f)$ on an
(infinite-dimensional) Gromov $\delta$-hyperbolic space (see Blanc-Cantat [8],
Theorem 4.4). Some other length functions are studied by Blanc and Furter [9]
for groups of birational maps, eg. dynamical number of base-points and
dynamical length.

##  3 Groups with purely positive length functions

###### Definition 3.1

A length function $l$ on a group $G$ is said to be purely positive if $l(g)>0$
for any infinite-order element $g\in G.$

In this section, we show that the (Gromov) hyperbolic group, mapping class
group and outer automorphism groups of free groups have purely positive length
functions. First, let us recall the relevant definitions.

A geodesic metric space $X$ is $\delta$-hyperbolic (for some real number
$\delta>0$) if for any geodesic triangle $\Delta xyz$ in $X,$ one side is
contained the $\delta$-neighborhood of the other two sides. A group $G$ is
(Gromov) hyperbolic if $G$ acts properly discontinuously and cocompactly on a
$\delta$-hyperbolic space $X$.

###### Definition 3.2

  1. (i)

An element $g$ in a group $G$ is called primitive if it cannot be writen as a
proper power $\alpha^{n},$ where $\alpha\in G$ and $|n|\geq 2;$

  2. (ii)

A group $G$ has unique-root property if every infinite-order element $g$ is a
proper power of a unique (up to sign) primitive element, i.e.
$g=\gamma^{n}=\gamma_{1}^{m}$ for primitive elements $\gamma,\gamma_{1}$ will
imply $\gamma=\gamma^{\pm}.$

The following fact is well-known.

###### Lemma 3.3

A torsion-free hyperbolic group has unique-root property.

Proof. Let $G$ be a torsion-free hyperbolic group and $1\neq g\in G.$ Suppose
that $g=\gamma^{n}=\gamma_{1}^{m}$ for primitive elements $\gamma$ and
$\gamma_{1}.$ The set $C_{G}(g)$ of centralizers is virtually cyclic (cf.
[16], Corollary 3.10, page 462). By a result of Serre, a torsion-free
virtually free group is free. Since $G$ is torsion-free, the group $C_{G}(g)$
is thus free and thus cyclic, say generated by $t$. Since $\gamma$ and
$\gamma^{\prime}$ are primitive, they are $t^{\pm}.$

###### Remark 3.4

The previous lemma does not hold for general hyperbolic groups with torsions.
For example, let $G=\mathbb{Z}/2\times\mathbb{Z}.$ We have
$(0,2)=(0,1)^{2}=(1,1)^{2}$ and $(0,1),$ $(1,1)$ are both primitive.

For a group $G,$ let $P(G)$ be the set of all primitive elements. We call two
primitive elements $\gamma,\gamma^{\prime}$ are general conjugate if there
exists $g\in G$ such that $g\gamma g^{-1}=\gamma^{\prime}$ or
$g\gamma^{-1}g=\gamma^{\prime}$. Let $\mathrm{CP}(G)$ be the general conjugacy
classes of primitive elements. For a set $S,$ let $S_{\mathbb{R}}$ be the set
of all real functions on $S$. The convex polyhedral cone spanned by $S$ is the
subset

$\\{\mathop{\textstyle\sum}\nolimits_{s\in S}a_{s}s\mid a_{s}\geq 0\
\\}\subset S_{\mathbb{R}}.$

###### Lemma 3.5

Let $G$ be a torsion-free hyperbolic group. The set of all length functions on
$G$ is the convex polyhedral cone spanned by the general conjugacy classes
$\mathrm{CP}(G).$

Proof. Let $l$ be a length function on $G.$ Then $l$ gives an element
$\mathop{\textstyle\sum}\nolimits_{s\in S}a_{s}s$ in the convex polyhedral
cone by $a_{s}=l(s).$ Conversely, for any general conjugacy classes
$[s]\in\mathrm{CP}(G)$ with $s$ a primitive element, let $l_{s}$ be the
function defined by $l_{s}(s^{\pm})=1$ and $l_{s}(\gamma)=0$ for element
$\gamma$ in any other general conjugacy classes. For any $1\neq g\in G,$ there
is a unique (up to sign) primitive element $\gamma$ such that $g=\gamma^{n}.$
Define $l_{s}(g)=|n|l_{s}(\gamma).$ Then $l_{s}$ satisfies conditions (1) and
(2) in Definition 1.1. The condition (3) is satisfied automatically, since any
commuting pair of elements $a,b$ generate a cyclic group in a torsion-free
hyperbolic group. Any element $\mathop{\textstyle\sum}\nolimits_{s\in
S}a_{s}s$ gives a length function on $G$ as a combination of $a_{s}l_{s}$.

###### Lemma 3.6

Let $G\ $be one of the following groups:

  * •

automorphism group $\mathrm{Aut}(F_{k})$ of a free group;

  * •

outer automorphism group $\mathrm{Out}(F_{k})$ of a free group or

  * •

mapping class group $\mathrm{MCG}(\Sigma_{g,m})$ (where $\Sigma_{g,m}$ is an
oriented surface of genus $g$ and $m$ punctures);

  * •

a hyperbolic group,

  * •

a $\mathrm{CAT}(0)$ group or more generally

  * •

a semi-hyperbolic group,

  * •

a group acting properly semi-simply on a $\mathrm{CAT}(0)$ space,

  * •

a group acting properly semi-simply on a $\delta$-hyperbolic space.

Then $G$ has a purely positive length function.

Proof. Note that hyperbolic groups and $\mathrm{CAT}(0)$ groups are
semihyperbolic (see [16], Prop. 4.6 and Cor. 4.8, Chapter III.$\Gamma$). For a
semihyperbolic group $G$ acting a metric space $X$ (actually $X=G$), the
translation $\tau$ is a length function by Lemma 2.7. Moreover, for any
infinite-order element $g\in G,$ the length $\tau(g)>0$ (cf. [16], Lemma 4.18,
page 479). For group acting properly semisimply on a $\mathrm{CAT}(0)$ space
(or a $\delta$-hyperbolic space), the translation

$l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}$

is a length function (cf. Lemma 2.7). For any hyperbolic $\gamma$, we get
$l(\gamma)>0.$ For any elliptic $\gamma,$ it is finite-order since the action
is proper.

Alibegovic [1] proves that the stable word length of
$\mathrm{Aut}(F_{n}),\mathrm{Out}(F_{n})$ are purely positive. Farb, Lubotzky
and Minsky [25] prove that Dehn twists and more generally all elements of
infinite order in $\mathrm{MCG}(\Sigma_{g,m})$ have positive translation
length.

###### Definition 3.7

A group $G$ is called poly-positive (or has a poly-positive length), if there
is a subnormal series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=G$

such that every finitely generated subgroup of the quotient $H_{i}/H_{i+1}$
$(i=0,...,n-1)$ has a purely positive length function.

Recall that a group $G$ is poly-free, if there is a subnormal series
$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=G$ such that the successive quotient $H_{i}/H_{i+1}$ is free
$(i=0,...,n-1).$ Since a free group is hyperbolic, it has a purely positive
length function. This implies that a poly-free group is poly-positive. A group
is said to have a virtual property if there is a finite-index subgroup has the
property.

Let $\Sigma$ be a closed oriented surfaec endowed with an area form $\omega.$
Denote by $\mathrm{Diff}(\Sigma,\omega)$ the group of diffeomorphisms
preserving $\omega$ and $\mathrm{Diff}_{0}(\Sigma,\omega)$ the subgroup
consisting of diffeomorphisms isotopic to the identity.

###### Lemma 3.8

When the genuse of $\Sigma$ is greater than $1,$ the group
$\mathrm{Diff}_{0}(\Sigma,\omega)$ and $\mathrm{Diff}(\Sigma,\omega)$ is poly-
positive.

Proof. This is eseentially proved by Py [54] (Section 1). There is a group
homomorphism

$\alpha:\mathrm{Diff}_{0}(\Sigma,\omega)\rightarrow H_{1}(\Sigma,\mathbb{R})$

with $\ker\alpha=\mathrm{Ham}(\Sigma,\omega)$ the group of Hamiltonian
diffeomorphisms of $\Sigma.$ Polterovich [53] (1.6.C.) proves that any
finitely generated group of $\mathrm{Ham}(\Sigma,\omega)$ has a purely
positive stable word length. Since the quotient group
$\mathrm{Diff}(\Sigma,\omega)/\mathrm{Diff}_{0}(\Sigma,\omega)$ is a subgroup
of the mapping class group $\mathrm{MCG}(\Sigma),$ which has a purely positive
stable word length by Farb-Lubotzky-Minsky [25], the group
$\mathrm{Diff}(\Sigma,\omega)$ is poly-positive.

##  4 Vanishing of length functions on abelian-by-cyclic groups

We will need the following result proved in [28].

###### Lemma 4.1

Given a group $G$, let $l:G\rightarrow[0,+\infty)$ be function such that

1) $l(e)=0;$

2) $l(x^{n})=|n|l(x)$ for any $x\in G,$ any $n\in\mathbb{Z};$

3) $l(xy)\leq l(x)+l(y)$ for any $x,y\in G.$

Then there exist a real Banach space $(\mathbb{B},\|\|)$ and a group
homomorphism $\varphi:G\rightarrow\mathbb{B}$ such that $l(x)=\|\varphi(x)\|$
for all $x\in G.$ Further more, if $l(x)>0$ for any $x\neq e$, one can take
$\varphi$ to be injective, i.e., an isometric embedding.

Let $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ be an abelian by cyclic group, where
$A=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in\mathrm{GL}_{2}(\mathbb{Z})$. We
prove Theorem 0.1 by proving the following two theorems.

###### Theorem 4.2

When the absolute value of the trace $|\mathrm{tr}(A)|>2,$ any length function
$l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes
on $\mathbb{Z}^{2}.$

Proof. Let $A=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\mathrm{GL}_{2}(\mathbb{Z}).$ Suppose that $t$ is a
generator of $\mathbb{Z}$ and

$t\begin{bmatrix}x\\\ y\end{bmatrix}t^{-1}=A\begin{bmatrix}x\\\
y\end{bmatrix}$

for any $x,y\in\mathbb{Z}.$ Note that

$(0,t^{k})(v,0)(0,t^{k})^{-1}=(A^{k}v,0)$

for any $v\in\mathbb{Z}^{2}$ and $k\in\mathbb{Z}$. Therefore, an element
$v\in\mathbb{Z}^{2}$ is conjugate to $A^{k}v$ for any integer $k.$ Note that

$A\begin{bmatrix}1\\\ 0\end{bmatrix}=\begin{bmatrix}a\\\
c\end{bmatrix},A^{2}\begin{bmatrix}1\\\
0\end{bmatrix}=\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}$

and

$\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}=(a+d)\begin{bmatrix}a\\\
c\end{bmatrix}-(ad-bc)\begin{bmatrix}1\\\ 0\end{bmatrix}.$

Therefore, we have

$\displaystyle|a+d|l(\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle=$
$\displaystyle l((a+d)\begin{bmatrix}a\\\ c\end{bmatrix})$ $\displaystyle=$
$\displaystyle l(\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}+(ad-
bc)\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle\leq$
$\displaystyle(1+|ad-bc|)l(\begin{bmatrix}1\\\ 0\end{bmatrix}).$

When $ad-bc=\pm 1$ and $|a+d|>2,$ we must have $l(\begin{bmatrix}1\\\
0\end{bmatrix})=0.$ Similarly, we can prove that $l(\begin{bmatrix}0\\\
1\end{bmatrix})=0.$ Since $l$ is subadditive on $\mathbb{Z}^{2},$ we get that
$l$ vanishes on $\mathbb{Z}^{2}.$

###### Theorem 4.3

When the absolute value $|\mathrm{tr}(A)|=|a+d|=2,I_{2}\neq
A\in\mathrm{SL}_{2}(\mathbb{Z}),$ any length function
$l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes
on the direct summand of $\mathbb{Z}^{2}$ spanned by eigenvectors of $A$.

Proof. We may assume that $A=\begin{bmatrix}1&n\\\ 0&1\end{bmatrix},n\neq 0.$
For any integer $k\geq 0$ and $v\in\mathbb{Z}^{2},$ we have

$t^{k}vt^{-k}=A^{k}v.$

Take $v=\begin{bmatrix}0\\\ 1\end{bmatrix}$ to get that

$t^{k}\begin{bmatrix}0\\\ 1\end{bmatrix}t^{-k}=\begin{bmatrix}kn\\\
0\end{bmatrix}+\begin{bmatrix}0\\\ 1\end{bmatrix}.$

Since the function $l|_{\mathbb{Z}^{2}}$ is given by the norm of a Banach
space according to Lemma 4.1, we get that

$\displaystyle k|n|l(\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle\leq$
$\displaystyle l(t^{k}\begin{bmatrix}0\\\
1\end{bmatrix}t^{-k})+l(\begin{bmatrix}0\\\ 1\end{bmatrix})$ $\displaystyle=$
$\displaystyle 2l(\begin{bmatrix}0\\\ 1\end{bmatrix}).$

Since $k$ is arbitrary, we get that $l(\begin{bmatrix}1\\\ 0\end{bmatrix})=0.$

###### Remark 4.4

When $A=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},$ the semidirect product
$G=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ is a Heisenberg group. A length
function on $G/Z(G)\cong\mathbb{Z}^{2}$ gives a length function on $G.$ In
particular, a length function of $G$ may not vanish on the second component
$\begin{bmatrix}0\\\ 1\end{bmatrix}\in\mathbb{Z}^{2}<G.$

###### Remark 4.5

When $A\in\mathrm{SL}_{2}(\mathbb{Z})$ has $|\mathrm{tr}(A)|<2,$ the matrix
$A$ is of finite order and the semi-direct product
$\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ contains $\mathbb{Z}^{3}$ as a finite-
index normal subgroup. Actually, in this case the group
$\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ is the fundamental group of a flat
$3$-manifold $M$ (see [61], Theorem 3.5.5). Therefore, the group
$\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ acts freely properly discontinuously
isometrically and cocompactly on the universal cover
$\tilde{M}=\mathbb{R}^{3}.$ This means the translation length gives a purely
positive length function on $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$.

###### Lemma 4.6

Let $A\in\mathrm{GL}_{n}(\mathbb{Z})$ be a matrix and
$G=\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$ the semi-direct product. Let
$\sum_{i=0}^{n}a_{i}x^{i}$ be the characterisitic polynomial of some power
$A^{k}$. Suppose that for some $k,$ there is a coefficient $a_{i}$ such that

$|a_{i}|>\sum_{j\neq i}|a_{j}|.$

Any length function $l$ of $G$ vanishes on $\mathbb{Z}^{n}.$

Proof. Let $t$ be a generator of $Z$ and $tat^{-1}=Aa$ for any $a\in Z^{n}.$
Note that for any integer $m,$ we have $t^{m}at^{-m}=A^{m}a$ and
$l(a)=l(A^{m}a).$ Note that

$\sum_{i=0}^{n}a_{i}A^{ki}=0$

and thus

$\displaystyle\sum_{i=0}^{n}a_{i}A^{ki}a$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle l(\sum_{i=0}^{n}a_{i}A^{ki}a)$ $\displaystyle=$ $\displaystyle
0$

for any $a\in\mathbb{Z}^{n}.$ Therefore,

$|a_{i}|l(a)=|a_{i}|l(A^{ki}a)=l(\sum_{j\neq i}a_{j}A^{kj}a)\leq\sum_{j\neq
i}|a_{j}|l(a).$

This implies that $l(a)=0.$

Proof of Corollary 0.2. When the group action is $C^{2},$ define

$l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\sup_{x\in
M}\|D_{x}f^{n}\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\sup_{x\in
M}\|D_{x}f^{-n}\|}{n}\\}$

for any diffeomorphism $f:M\rightarrow M.$ Lemma 2.5 shows that $l$ is a
length function, which is an upper bound of the Lyapunov exponents. When the
group action is Lipschitz, define

$L(f)=\sup_{x\neq y}\frac{d(fx,fy)}{d(x,y)}$

for a Lipschitz-homeomorphism $f:M\rightarrow M.$ Since $L(fg)\leq L(f)L(g)$
for two Lipschitz-homeomorphisms $f,g:M\rightarrow M,$ we have that
$l(f):=\lim_{n\rightarrow\infty}\frac{\max\\{\log(L(f^{n})),\log(L(f^{-n}))\\}}{n}$
gives a length function by Lemma 2.1. Note that $l(f)\geq h_{top}(f)$ (see
[40], Theorem 3.2.9, page 124). The vanishings of the topological entropy
$h_{top}$ and the Lyapunov exponents in Corolary 0.2 are proved by Theorem 0.1
considering these length functions.

##  5 Classification of length functions on nilpotent groups

The following lemma is a key step for our proof of the vanishing of length
functions on Heisenberg groups.

###### Lemma 5.1

Let

$G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$

be the Heisenberg group. Suppose that $f:G\rightarrow\mathbb{R}$ is a
conjugation-invariant function, i.e. $f(xgx^{-1})=f(g)$ for any $x,g\in G.$
For any coprime integers (not-all-zero) $m,n$ and any integer $k,$ we have

$f(a^{m}b^{n}c^{k})=f(a^{m}b^{n}).$

Proof. It is well-known that for any integers $n,m,$ we have
$[a^{n},b^{m}]=c^{nm}.$ Actually, since $aba^{-1}b^{-1}=c,$ we have
$ba^{-1}b^{-1}=a^{-1}c$ and thus $ba^{-n}b^{-1}=a^{-n}c^{n}$ for any integer
$n.$ Therefore, $a^{n}ba^{-n}b^{-1}=c^{n}$ and $a^{n}ba^{-n}=c^{n}b,$
$a^{n}b^{m}a^{-n}=a^{nm}b^{m}$ for any integer $m.$ This means
$[a^{n},b^{m}]=c^{nm}.$ For any coprime $m,n$, and any integer $k,$ let
$s,t\in\mathbb{Z}$ such that $ms+nt=k.$ We have

$a^{-m}b^{-s}a^{m}b^{s}=c^{ms},b^{-s}a^{m}b^{s}=a^{m}c^{ms},b^{-s}a^{m}b^{n}b^{s}=a^{m}b^{n}c^{ms}$

and

$a^{t}b^{n}a^{-t}b^{-n}=c^{nt},a^{t}b^{n}a^{-t}=b^{n}c^{nt},a^{t}a^{m}b^{n}a^{-t}=a^{m}b^{n}c^{nt}.$

Therefore,

$a^{t}(b^{-s}a^{m}b^{n}b^{s})a^{-t}=a^{t}(a^{m}b^{n}c^{ms})a^{-t}=a^{m}b^{n}c^{nt+ms}=a^{m}b^{n}c^{k}.$

When $f$ is conjugation-invariant, we get $f(a^{m}b^{n}c^{k})=f(a^{m}b^{n})$
for any coprime $m,n$, and any integer $k.$

###### Lemma 5.2

Let

$G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$

be the Heisenberg group. Any length function $l:G\rightarrow[0,\infty)$ (in
the sense of Definition 1.1) factors through the abelization
$G_{\mathrm{ab}}:=G/[G,G]\cong\mathbb{Z}^{2}.$ In other words, there is a
function

$l^{\prime}:G_{\mathrm{ab}}\rightarrow[0,\infty)$

such that $l^{\prime}(x^{n})=|n|l^{\prime}(x)$ for any $x\in G_{\mathrm{ab}},$
any integer $n$ and $l=l^{\prime}\circ q$, where $q:G\rightarrow
G_{\mathrm{ab}}$ is the natural quotient group homomorphism.

Proof. Let $H=\langle c\rangle\cong\mathbb{Z}$ and write $G=\cup gH$ the union
of left cosets. We choose the representative $g_{ij}=a^{i}b^{j}$ with
$(i,j)\in\mathbb{Z}^{2}.$ Note that the subgroup $\langle g_{ij},c\rangle$
generated by $g_{ij},c$ is isomorphic to $\mathbb{Z}^{2}$ for coprime $i,j.$
The length function $l$ is subadditive on $\langle g_{ij},c\rangle.$ By Lemma
4.1, there is a Banach space $\mathbb{B}$ and a group homomorphism $\varphi:$
$\langle g_{ij},c\rangle\rightarrow\mathbb{B}$ such that $l(g)=\|\varphi(g)\|$
for any $g\in$ $\langle g_{ij},c\rangle.$ Lemma 5.1 implies that

$\|\varphi(g_{ij})+\varphi(c^{k})\|=\|\varphi(g_{ij})\|$

for any integer $k.$ Since

$\|\varphi(g_{ij})\|=\|\varphi(g_{ij})+\varphi(c^{k})\|\geq|k|\|\varphi(c)\|-\|\varphi(g_{ij})\|$

for any $k,$ we have that $\|\varphi(c)\|=l(c)=0.$ This implies that

$l(g_{ij}^{n}c^{m})=\|\varphi(g_{ij}^{n})+\varphi(c^{m})\|=l(g_{ij}^{n})$

for any integers $m,n.$ Moreover, for any integers $m,n$ and coprime $i,j,$ we
have $a^{ni}b^{nj}c^{m}=(a^{i}b^{j})^{n}c^{k}$ for some integer $k.$ This
implies that

$l(a^{ni}b^{nj}c^{m})=l((a^{i}b^{j})^{n})=|n|l(a^{i}b^{j}).$

Therefore, the function $l$ is constant on each coset $gH.$ Define
$l^{\prime}(gH)=l(g).$ Since $l(g^{k})=|l|l(g),$ we have that
$l^{\prime}(g^{k}H)=|k|l^{\prime}(gH).$ The proof is finished.

Denote by $S^{\prime}=\\{(m,n)\mid m,n$ are coprime integers$\\}$ be the set
of coprime integer pairs and define an equivalence relation by
$(m,n)\sim(m^{\prime},n^{\prime})$ if $(m,n)=\pm(m^{\prime},n^{\prime})$. Let
$S=S^{\prime}/\sim$ be the equivalence classes.

###### Theorem 5.3

Let $G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$ be the
Heisenberg group. The set of all length functions $l:G\rightarrow[0,\infty)$
(in the sense of Definition 1.1) is the convex polyhedral cone

$\mathbb{R}_{\geq 0}[S]=\\{\sum_{s\in S}a_{s}s\mid a_{s}\in\mathbb{R}_{\geq
0},s\in S\\}.$

Proof. Similar to the proof of the previous lemma, we let $H=\langle
c\rangle\cong\mathbb{Z}$ and write $G=\cup gH$ the union of left cosets. We
choose the representative $g_{ij}=a^{i}b^{j}$ with $(i,j)\in\mathbb{Z}^{2}.$
Let $T$ be the set of length function $l:G\rightarrow[0,\infty).$ For any
length functions $l,$ let $\varphi(l)=\sum_{s\in S}a_{s}s\in\mathbb{R}_{\geq
0}[S],$ where $a_{s}=l(a^{i}b^{j})$ with $(i,j)$ a representative of $s.$ Note
that

$l(a^{-i}b^{-j})=l(b^{-j}a^{-i}c^{ij})=l(b^{-j}a^{-i})=l(a^{i}b^{j}),$

which implies that $a_{s}$ is well-defined. We have defined a function
$\varphi:T\rightarrow\mathbb{R}_{\geq 0}[S].$ If
$\varphi(l_{1})=\varphi(l_{2})$ for two functions $l_{1},l_{2},$ then
$l_{1}(a^{i}b^{j})=l_{2}(a^{i}b^{j})$ for coprime integers $i,j.$ Since both
$l_{1},l_{2}$ are conjugation-invariant, Lemma 5.1 implies that $l_{1},l_{2}$
coincide on any coset $a^{i}b^{j}H$ and thus on the whole group $G.$ This
proves the injectivity of $\varphi.$ For any $\sum_{s\in S}a_{s}s,$ we define
a function

$l:G=\cup a^{i}b^{j}H\rightarrow[0,\infty).$

For any coprime integers $i,j,$ define $l(a^{i}b^{j}z)=a_{s}$ for any
representative $(i,j)$ of $s$ and any $z\in H.$ For any general integers $m,n$
and $z\in H,$ define
$l(a^{m}b^{n}z)=l(a^{m}b^{n})=|\gcd(m,n)|l(a^{m/\gcd(m,n)}b^{n/\gcd(m,n)})$
and $l(z)=0.$ From the definition, it is obvious that $l$ is homogeous. Note
that any element of $G$ is of the form $a^{k}b^{s}c^{t}$ for integers
$k,s,t\in\mathbb{Z}.$ For any two elements
$a^{k}b^{s}c^{t},a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}}$ we have the
conjugation

$a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}}a^{k}b^{s}c^{t}(a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}})^{-1}=a^{k}b^{s}c^{t^{\prime\prime}}$

for some $t^{\prime\prime}\in\mathbb{Z}.$ Therefore, we see that $l$ is
conjugation-invariant. The previous equality also shows that two elements
$g,h$ are commuting if and only if they lies simutanously in $\langle
a^{i}b^{j},c\rangle$ for a pair of coprime integers $i,j.$ By construction, we
have $l(g)=l(h).$ This proves the surjectivity of $\varphi.$

##  6 Length functions on matrix groups

In this section, we study length functions on matrix groups
$\mathrm{SL}_{n}(\mathbb{R})$. As the proofs are elementary, we present here
in a separated section, without using profound results on Lie groups and
algebraic groups. The following lemma is obvious.

###### Lemma 6.1

Let $G_{p,q}=\langle x,t:tx^{p}t^{-1}=x^{q}\rangle$ be a Baumslag-Solitar
group. When $|p|\neq|q|,$ any length function $l$ on $G$ has $l(x)=0.$

Proof. Note that $|p|l(x)=l(x^{p})=l(x^{q})=|q|l(x),$ which implies $l(x)=0.$

Let $V^{n}$ be a finite-dimensional vector space over a field $K$ and
$A:V\rightarrow V$ a unipotent linear transformation (i.e. $A^{k}=0$ for some
positive integer $k).$ The following fact is from linear algebra (see the
Lemma of page 313 in [4]. Since the reference is in Chinese, we repeat the
proof here).

###### Lemma 6.2

$I+A$ is conjugate to a direct sum of Jordan blocks with $1s$ along the
diagonal.

Proof. We prove that $V$ has a basis

$\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}$

satisfying $A^{k_{i}}a_{i}=0$ for each $i,$ which implies that the
representation matrix of $I+A$ is similar to a direct sum of Jordan blocks
with $1$ along the diagonal. The proof is based on the induction of $\dim V.$
When $\dim V=1,$ choose $0\neq v\in V.$ Suppose that $Av=\lambda v.$ Then
$A^{k}v=\lambda^{k}v=0$ and thus $\lambda=0.$ Suppose that the case is proved
for vector spaces of dimension $k<n.$ Note that the invariant subspace $AV$ is
a proper subspace of $V$ (otherwise, $AV=V$ implies $A^{k}V=A^{k-1}V=V=0$). By
induction, the subspace $AV$ has a basis

$\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}.$

Choose $b_{i}\in V$ satisfying $A(b_{i})=a_{i}.$ Then $A$ maps the set

$\\{b_{1},Ab_{1},\cdots,A^{k_{1}}b_{1},b_{2},Ab_{2},\cdots,A^{k_{2}}b_{2},b_{s},\cdots,Ab_{s},\cdots,A^{k_{s}}b_{s}\\}$

to the basis

$\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}.$

This implies that the former set is linearly independent (noting that
$A(A^{k_{i}}b_{i})=0$). Extend this set to be a $V^{\prime}s$ basis

$\\{b_{1},Ab_{1},\cdots,A^{k_{1}-1}b_{1},b_{2},Ab_{2},\cdots,A^{k_{2}-1}b_{2},b_{s},\cdots,Ab_{s},\cdots,A^{k_{s}-1}b_{s},b_{s+1},\cdots,b_{s^{\prime}}\\}.$

Note that $Ab_{i}=0$ for $i\geq s+1$ and $A^{k_{i}+1}b_{i}=A^{k_{i}}a_{i}=0$
for each $i\leq s.$ This finishes the proof.

###### Corollary 6.3

Let $A_{n\times n}$ be a strictly upper triangular matrix over a field $K$ of
characteristic $\mathrm{ch}(K)\neq 2$. Then $A^{2}$ is conjugate to $A.$

Proof. Suppose that $A=I+u$ for a nilpotent matrix $u.$ Lemma 6.2 implies that
$A^{2}$ is conjugate to a direct sum of Jordan blocks. Without loss of
generality, we assume $A$ is a Jordan block. Then $A^{2}=I+2u+u^{2}.$ By Lemma
6.2 again, $A^{2}$ is conjugate to a direct sum of Jordan blocks with $1s$
along the diagonal. The minimal polynomial of $A^{2}$ is $(x-1)^{n},$ which
shows that there is only one block in the direct sum and thus $A^{2}$ is
conjugate to $A.$

Recall that a matrix $A\in\mathrm{GL}_{n}(\mathbb{R})$ is called semisimple if
as a complex matrix $A$ is conjugate to a diagonal matrix. A semisimple matrix
$A$ is elliptic (respectively, hyperbolic) if all its (complex) eigenvalues
have modulus $1$ (respectively, are $>0$). The following lemma is the complete
multiplicative Jordan (or Jordan-Chevalley) decomposition (cf. [34], Lemma
7.1, page 430).

###### Lemma 6.4

Each $A\in$ $\mathrm{GL}_{n}(\mathbb{R})$ can be uniquely writen as $A=ehu,$
where $e,h,u\in\mathrm{GL}_{n}(\mathbb{R})$ are elliptic, hyperbolic and
unipotent, respectively, and all three commute.

The following result characterizes the continuous length functions on compact
Lie groups.

###### Lemma 6.5

Let $G$ be a compact connected Lie group and $l$ a continuous length function
on $G$. Then $l=0.$

Proof. For any element $g\in G,$ there is a maximal torus $T\backepsilon g.$
For finite order element $h\in T,$ we have $l(h)=0.$ Note that the set of
finite-order elements is dense in $T.$ Since $l$ is continuous, $l$ vanishes
on $T$ and thus $l(g)=0$ for any $g.$

###### Theorem 6.6

Let $G=\mathrm{SL}_{n}(\mathbb{R})$ $(n\geq 2).$ Let
$l:G\rightarrow[0,+\infty)$ be a length function, which is continuous on
compact subgroups and the subgroup of diagonal matrices with positive diagonal
entries. Then $l$ is uniquely determined by its images on the subgroup $D$ of
diagonal matrices with positive diagonal entries.

Proof. For any $g\in\mathrm{SL}_{n}(\mathbb{R}),$ let $g=ehu$ be the Jordan
decomposition for commuting elements $e,h,u$, where $e$ is elliptic, $h$ is
hyperbolic and $u$ is unipotent (see Lemma 6.4) after multiplications by
suitable powers of derterminants. Then

$l(g)\leq l(e)+l(h)+l(u).$

For any unipotent matrix $u,$ there is an invertible matrix $a$ such that
$aua^{-1}$ is strictly upper triangular (see [34], Theorem 7.2, page 431).
Lemma 6.3 implies that $u^{2}$ is conjugate to $u$. Therefore, $l(u)=0$ by
Lemma 6.1. Since $l$ vainishes on a compact Lie group (cf. Lemma 6.5), we have
that $l(e)=0$ for any elliptic matrix $e.$ Therefore, $l(g)\leq l(h).$
Similarly,

$l(h)=l(e^{-1}gu^{-1})\leq l(g)$

which implies $l(g)=l(h).$ Note that a hyperbolic matrix is conjugate to a
real diagonal matrix with positive diagonal entries.

Proof of Theorem 0.10. By Theorem 6.6, the length function $l$ is determined
by its image on the subgroup $D$ generated by $h_{12}(x),x\in\mathbb{R}_{>0}.$
Take $x=e^{t},t\in\mathbb{R}$. We have
$l(h_{12}(e^{\frac{k}{l}}))=\frac{|k|}{|l|}l(h_{12}(e))$ for any rational
number $\frac{k}{l}.$ Since $l$ satisfies the condition 2) of the definition
and is continuous on $D$, we see that $l|_{D}$ is determined by the image
$l(h_{12}(e))$ (actually, any real number $t$ is a limit of a rational
sequence). Note that the translation function $\tau$ vainishes on compact
subgroups and is continuous on the subgroup of diagonal subgroups with
positive diagonal entries (cf. [16], Cor. 10.42 and Ex. 10.43, page 320).
Therefore, $l$ is proposional to $\tau.$ Actually, $\tau$ can be determined
explicitly by the formula $\mathrm{tr}(A)=\pm 2$cosh$\frac{\tau(A)}{2}$ (for
nonzero $\tau(A)$), where $\mathrm{tr}$ is the trace and cosh is the
hyperbolic cosine function (see [5], Section 7.34, page 173). This implies
that $l(A)$ is determined by the spectrum radius of $A$ (which could also be
seen clearly by the matrix norm).

Let $h_{1i}(x)$ ($i=2,\cdots,n$) be an $n\times n$ diagonal matrix whose
$(1,1)$-entry is $x,$ $(i,i)$-entry is $x^{-1},$ while other diagonal entries
are $1$s and non-diagonal entries are $0$s. The subgroup
$D<\mathrm{SL}_{n}(\mathbb{R})$ of diagonal matrices with positive diagonal
entries is isomorphic to $(\mathbb{R}_{>0})^{n-1}$ and $D$ is generated by the
matrices $h_{1i}(x)$ ($i=2,\cdots,n$) whose $(1,1)$-entry is $x,$
$(i,i)$-entry is $x^{-1}.$ Since $h_{1i}(x)$ $(i\neq 1)$ is conjugate to
$h_{12}(x),$ a length function
$l:\mathrm{SL}_{n}(\mathbb{R})\rightarrow[0,+\infty)$ is completely determined
by its image on the convex hull spanned by
$h_{12}(e),h_{13}(e),\cdots,h_{1n}(e)$ (see Theorem 7.10 for a more general
result on Lie groups). Here $e$ is the Euler’s number in the natural
exponential function.

###### Corollary 6.7

Let $l:\mathrm{SL}_{2}(\mathbb{R})\rightarrow[0,+\infty)$ be a non-trivial
length function that is continuous on the subgroup $SO(2)$ and the diagonal
subgroup. Then $l(g)>0$ if and only if $g$ is hyperbolic.

Proof. It is well-known that the elements in $\mathrm{SL}_{2}(\mathbb{R})$ are
classified as elliptic, hyperbolic and parabolic elements. Moreover, the
translation length $\tau$ vanishes on the compact subgroup $SO(2)$ and the
parabolic elements. The corollary follows Theorem 0.10.

When the length function $l$ is the asymptotic distortion function
$\mathrm{dist}_{\infty}$, Corollary 6.7 is known to Navas [24] (Proposition
4).

##  7 Length functions on algebraic and Lie groups

For an algebraic group $G,$ let $k[G]$ be the regular ring. For any $g\in G,$
let $\rho_{g}:k[G]\rightarrow k[G]$ be the right translation by $x.$ The
following is the famous Jordan (or Jordan-Chevalley) decomposition.

###### Lemma 7.1

([37] p.99) Let $G$ be an algebraic group and $g\in G.$ There exists unique
elements $g_{s},g_{u}$ such that $g_{s}g_{u}=g_{u}g_{s},$ and $\rho_{g_{s}}$
is semisimple, $\rho_{g_{u}}$ is unipotent.

###### Lemma 7.2

[46] Let $G$ be a reductive connected algebraic group over an algebraically
closed field $k.$ The conjugacy classes of unipotent elements in $G$ is
finite.

###### Lemma 7.3

([29], Theorem 3.4) If $G$ is a reductive linear algebraic group defined over
a field $k$ and $g\in G(k)$ then the set of conjugacy classes in $G(k)$ which
when base changed to the algebraic closed field $\bar{k}$ are equal to the
conjugacy class of $g$ in $G(\bar{k})$ is in bijection with the subset of
$H^{1}(\bar{k}/k,Z(g)(k)),$ the Galois cohomology group.

###### Definition 7.4

A field $k$ is of type (F) if for any integer $n$ there exist only finitely
many extensions of $k$ of degree $n$ (in a fixed algebraic closure $\bar{k}$
of $k$).

Examples of fields of type (F) include: the field $\mathbb{R}$ of reals, a
finite field, the field of formal power series over an algebraically closed
field.

###### Lemma 7.5

[Borel-Serre [12], Theorem 6.2] Let $k$ be a field of type (F) and let $H$ be
a linear algebraic group defined over $k$. The set $H^{1}(\bar{k}/k,H(k))$ is
finite.

###### Lemma 7.6

Let $G(k)$ be a reductive linear algebraic group over a field of type (F) and
$l$ a length function on $G.$ Then $l(g)=l(g_{s}),$ where $g_{s}$ is the
semisimple part of $g.$

Proof. By the Jordan decomposition $g=g_{s}g_{u},$ we have $l(g)\leq
l(g_{s})+l(g_{u})$ and $l(g_{s})\leq l(g)+l(g_{u}^{-1}).$ Note that for any
integer $n,$ $g_{u}^{n}$ is also unipotent. By the Lemma 7.2, Lemma 7.3 and
Lemma 7.5, there are only finitely many conjugacy classes of unipotent
elements. This implies that $g_{u}^{n_{1}}=g_{u}^{n_{2}}$ for distinct
positive integers $n_{1},n_{2}.$ Therefore, we have
$n_{1}l(g_{u})=n_{2}l(g_{u}),$ which implies that $l(g_{u})=0$ and thus
$l(g)=l(g_{s}).$

A Lie group $G$ is semisimple if its maximal connected solvable normal
subgroup is trivial. Let $\mathfrak{g}$ be its Lie algebra and let
$\exp:\mathfrak{g}\rightarrow G$ denote the exponential map. An element
$x\in\mathfrak{g}$ is real semi-simple if $Ad(x)$ is diagonalizable over
$\mathbb{R}$. An element $g\in G$ is called hyperbolic (resp. unipotent) if
$g$ is of the form $g=\exp(x)$ where $x$ is real semi-simple (resp.
nilpotent). In either case the element $x$ is easily seen to be unique and we
write $x=\log g$. The following is the Jordan decomposition in Lie groups. An
element $e\in G$ is elliptic if $Ad(e)$ is diagonalizable over $\mathbb{C}$
with eigenvalues $1$.

###### Lemma 7.7

([42], Prop. 2.1 and Remark 2.1)

  1. 1.

Let $g\in G$ be arbitrary. Then g may be uniquely written

$g=e(g)h(g)u(g)$

where $e(g)$ is elliptic, $h(g)$ is hyperbolic and $u(g)$ is unipotent and
where the three elements $e(g),h(g),u(g)$ commute.

  2. 2.

An element $f\in G$ commutes with $g$ if and only if $f$ commutes with the
three components. Moreover, if $f,g$ commutes, then

$e(fg)=e(f)e(g),h(fg)=h(f)h(g),u(fg)=u(f)u(g).$

###### Lemma 7.8

[Eberlein, Prop. 1.14.6, page 63]Let $G$ be a connected semisimple Lie group
whose center is trivial. Then there exists an integer $n\geq 2$ and an
algebraic group $G^{\ast}<\mathrm{GL}_{n}(\mathbb{C})$ defined over
$\mathbb{Q}$ such that $G$ is isomorphic to $G_{\mathbb{R}}^{\ast 0}$ (the
connected component of $G_{\mathbb{R}}^{\ast 0}$ containing the identity) as a
Lie group.

Let $G=KAN$ be an Iwasawa decomposition. The Weyl group $W$ is the finite
group defined as the quotient of the normalizer of $A$ in $K$ modulo the
centralizer of $A$ in $K.$ For an element $x\in A,$ let $W(h)$ be the set of
all elements in $A$ which are conjugate to $x$ in $G.$

###### Lemma 7.9

([42], Prop. 2.4) An element $h\in G$ is hyperbolic if and only if it is
conjugate to an element in $A.$ In such a case, $W(h)$ is a single $W$-orbit
in $A.$

###### Theorem 7.10

Let $G$ be a connected semisimple Lie group whose center is finite with an
Iwasawa decomposition $G=KAN$. Let $W$ be the Weyl group.

  1. (i)

Any length function $l$ on $G$ that is continuous on the maximal compact
subgroup $K$ is determined by its image on $A.$

  2. (ii)

Conversely, any length function $l$ on $A$ that is $W$-invariant (i.e.
$l(w\cdot a)=l(a)$) can be extended to be a length function on $G$ that
vanishes on the maximal compact subgroup $K.$

Proof. (i) Let $Z$ be the center of $G.$ Then $G/Z$ is connected with trivial
center. For any $z\in Z,g\in G,$ we have $l(z)=0$ and $l(gz)=l(g).$ The length
function $l$ factors through a length function on $G/Z.$ We may assume that
$G$ has the trivial center. For any $g\in G,$ the Jordan decomposition gives
$g=ehu,$ where $e$ is elliptic, $h$ is hyperbolic and $u$ is unipotent and
where the three elements $e,h,u$ commute (cf. Lemma 7.7). By Lemma 7.8, the
Lie group $G$ is an algebraic group. Lemma 7.6 implies that $l$ vanies on
unipotent elements and $l(g)=l(eh).$ Since $l$ vanishes on $e$ (cf. Lemma
6.5), we have $l(g)=l(h).$ Therefore, the function $l$ is determined by its
image on $A.$

(ii) Let $l$ be a length function $l$ on $A$ that is $W$-invariant. We first
extend $l$ to the set $H$ of all the conjugates of $A.$ For any $g\in G,a\in
A,$ define $l^{\prime}(gag^{-1})=l(a).$ If

$g_{1}a_{1}g_{1}^{-1}=g_{2}a_{2}g_{2}^{-1}$

for $g_{1},g_{2}\in G,a_{1},a_{2}\in A,$ then
$g_{1}^{-1}g_{2}a_{2}g_{2}^{-1}g_{1}=a_{1}.$ By Lemma 7.9, there exists an
element $w\in W$ such that $w\cdot a_{2}=a_{1}.$ Therefore, we have
$l(a_{1})=l(a_{2})$ and thus $l^{\prime}$ is well-defined on the set $H$ of
conjugates of elements in $A.$ Such a set $H$ is the set consisting of
hyperbolic elements by Lemma 7.9. We then extend $l^{\prime}$ on the set of
all conjugates of elements in $K.$ For any $g\in G,k\in K,$ define
$l^{\prime}(gkg^{-1})=0.$ If

$g_{1}kg_{1}^{-1}=g_{2}ag_{2}^{-1}$

for $g_{1},g_{2}\in G,k\in K,a\in A,$ then
$g_{1}^{-1}g_{2}ag_{2}^{-1}g_{1}=k.$ Then $k$ is both hyperbolic and elliptic.
The only element which is both elliptic and hyperbolic is the identity
element. Therefore, we have $k=a=1$ and

$l^{\prime}(g_{1}kg_{1}^{-1})=l(g_{2}ag_{2}^{-1})=1.$

This shows that $l^{\prime}$ is well-defined on the set of hyperbolic elements
and elliptic elements. For any unipotent element $u\in G,$ define
$l^{\prime}(u)=0.$ For any element $g\in G,$ let $g=ehu$ be the Jordan
decomposition. Define $l^{\prime}(g)=l^{\prime}(h).$

We check the function $l^{\prime}$ is a length function on $G.$ The definition
shows that $l^{\prime}$ is conjugate invariant. For any positive integer $n$
and any $g\in G$ with Jordan decomposition $g=ehu,$ we have
$g^{n}=e^{n}h^{n}u^{n}$ and thus $l^{\prime}(g^{n})=l^{\prime}(h^{n}).$ But
$h^{n}$ is hyperbolic and conjugate to an element in $A$ (see Lemma 7.9).
Therefore, we have $l^{\prime}(h^{n})=|n|l^{\prime}(h)$ and thus
$l^{\prime}(g^{n})=|n|l^{\prime}(g).$ If $g_{1}=e_{1}h_{1}u_{1}$ commutes with
$g_{2}=e_{2}h_{2}u_{2},$ then

$g_{1}g_{2}=e_{1}e_{2}h_{1}h_{2}u_{1}u_{2}$

(cf. Lemma 7.7) and $l^{\prime}(g_{1}g_{2})=l^{\prime}(h_{1}h_{2}).$ Since
$h_{1},h_{2}$ are commuting hyperbolic elements, they are conjuate
simutaniously to elements in $A.$ Therefore, we have

$l^{\prime}(h_{1}h_{2})\leq l^{\prime}(h_{1})+l^{\prime}(h_{2})$

and

$l^{\prime}(g_{1}g_{2})\leq l^{\prime}(g_{1})+l^{\prime}(g_{2}).$

###### Remark 7.11

A length function $l$ on $A$ is determined by a group homomorphism
$f:A\rightarrow\mathbb{B},$ for a real Banach space $(\mathbb{B},\|\|),$
satisfying $l(a)=\|f(a)\|$ (see Lemma 4.1). The previous theorem implies that
a length function $l$ on the Lie group $G$ (that is continuous on compact
subgroup) is uniqely determined by such a group homomorphism
$f:A\rightarrow\mathbb{B}$ such that $\|f(a)\|=\|f(wa)\|$ for any $a\in A$ and
$w\in W,$ the Weyl group.

Let $G$ be a connected semisimple Lie group whose center is finite with an
Iwasawa decomposition $G=KAN$. Let $\exp:\mathfrak{g}\rightarrow G$ be the
exponent map from the Lie algebra $\mathfrak{g}$ with subalgebra
$\mathfrak{h}$ corresponding to $A$.

###### Theorem 7.12

Suppose that $l$ is a length function on $G$ that is continuous on $K$ and
$A.$ Then $l$ is determined by its image on $\exp(v)$ $($unit vector
$v\in\mathfrak{h})$ in a fixed closed Weyl chamber of $A.$

Proof. Let $Z$ be the center of $G.$ Then $G/Z$ is connected with trivial
center. The length function $l$ factors through an length function on $G/Z$
(cf. Corrolary 1.8). For any $g\in G$ we have $g=ehu,$ where $e$ is elliptic,
$h$ is hyperbolic and $u$ is unipotent and where the three elements $e,h,u$
commute (cf. Lemma 7.7). By Lemma 7.8 and Lemma 7.6, we have $l(g)=l(eh).$
Since $l$ vanishes on $e$ (cf. Lemma 6.5), we have $l(g)=l(h).$ Any element
$h\in A$ is conjugate to an element in a fixed Weyl chamber $C$ (cf. [38],
Theorem 8.20, page 254). For any element $\exp(x)\in C,$ with unit vector
$x\in\mathfrak{h},$ the one-parameter subgroup $\exp(\mathbb{R}x)$ lies in
$A.$ Since $l$ is continuous on $A,$ the function $l$ is determined by its
image on $\exp(\mathbb{Q}x).$ Note that
$l(e^{\frac{m}{n}x})=\frac{1}{n}l(e^{mx})=\frac{m}{n}l(e^{x})$ for any
rational number $\frac{m}{n}.$ The function $l$ is determined by $l(e^{x}),$
for all unit vectors $x$ in the fixed closed Weyl chamber.

###### Corollary 7.13

Let $G$ be a connected semisimple Lie group whose center is finite of real
rank $1.$ There is essentially only one length function on $G.$ In order
words, any continuous length function is proportional to the translation
function on the symmetric space $G/K.$

Proof. When the real rank of $G$ is $1,$ a closed Weyl chamber is of dimension
$1$. Therefore, the previous theorem implies that any continuous length
function is determined by its image on a unit vector in a split torus.

##  8 Rigidity of group homomorphisms on arithmetic groups

Let $V$ denote a finite-dimensional vector space over $\mathbb{C}$, endowed
with a $\mathbb{Q}$-structure. Recall that the arithmetic subgroup is defined
as the following (cf. Borel [11], page 37).

###### Definition 8.1

Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GL}(V)$. A subgroup $\Gamma$ of
$G_{\mathbb{Q}}$ is said to be arithmetic if there exists a lattice $L$ of
$V_{\mathbb{Q}}$ such that $\Gamma$ is commensurable with $G_{L}=\\{g\in
G_{\mathbb{Q}}:gL=L\\}.$

###### Theorem 8.2

Let $\Gamma$ be an arithmetic subgroup of a simple algebraic
$\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2.$ Suppose that $H$ is a
group with a purely positive length function. Then any group homomorphism
$f:\Gamma\rightarrow H$ has its image finite.

Recall that a group $G$ is quasi-simple, if any non-trivial normal subgroup is
either finite or of finite index. The Margulis-Kazhdan theorem (see [62],
Theorem 8.1.2) implies that an irreducible lattice (and hence) in a semisimple
Lie group of real rank $\geq 2$ is quasi-simple.

###### Lemma 8.3

Let $\Gamma$ be a finitely generated quasi-simple group with contains a
Heisenberg subgroup, i.e. there are elements torsion-free elements
$a,b,c\in\Gamma$ satisfying $[a,b]=c,[a,c]=[c,b]=1$. Suppose that $G$ has a
virtually poly-positive length. Then any group homomorphism
$f:\Gamma\rightarrow G$ has its image finite.

Proof. Suppose that $G$ has a finite-index subgroup $H$ and a subnormal series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=H$

such that every finitely generated subgroup of $H_{i}/H_{i+1}$ has a purely
positive length function. Without loss of generality, we assume that $H$ is
normal. Let $f:\Gamma\rightarrow G$ be a homomorphism. The kernel of the
composite

$f_{0}:\Gamma\overset{f}{\rightarrow}G\rightarrow G/H$

is finitely generated. Suppose that the image of the composite

$f_{1}:\ker f_{0}\overset{f}{\rightarrow}H\rightarrow H/H_{1}$

has a purely positive length function $l.$ After passing to finite-index
subgroups, we may still suppose that $\ker f_{0}$ contains a Heisenberg
subgroup $\langle a,b,c\rangle.$ By Lemma 5.2, the length function $l$
vanishes on $f_{1}(c).$ Therefore $f_{1}(c^{k})=1\in H/H_{1}$ for some integer
$k>0.$ The normal subgroup $\ker f_{1}$ containing $c^{k}$ is of finite index.
Now we have map $\ker f_{1}\rightarrow H_{1}$ induced by $f.$ An induction
argument shows that $f$ maps some power $c^{d}$ of the central element of the
Heisenberg subgroup into the identity $1\in G.$ Therefore, the image of $f$ is
finite.

Proof of Theorem 8.2. It is well-known that that $G$ contains a
$\mathbb{Q}$-split simple subgroup whose root system is the reduced subsystem
of the $\mathbb{Q}$-root system of $G$ (see [13], Theorem 7.2, page 117).
Replacing $G$ with this $\mathbb{Q}$-subgroup, we may assume $G$ is
$\mathbb{Q}$-split and the root system of G is reduced. Because $G$ is simple
and $\mathbb{Q}$-rank($G$)$\geq 2$, we know that the $\mathbb{Q}$-root system
of G is irreducible and has rank at least two. Therefore, the
$\mathbb{Q}$-root system of $G$ contains an irreducible subsystem of rank two,
that is, a root subsystem of type $A_{2},B_{2},G_{2}$ (see [60], page 338).
For $A_{2},$ choose $\\{\alpha_{1},\alpha_{3}\\}$ as a set of simple roots
(see Figure 1).

Then the root element $x_{\alpha_{1}+\alpha_{3}}(rs)=x_{\alpha_{2}}(rs)$ is a
commutator $[x_{\alpha_{1}}(r),x_{\alpha_{3}}(s)],$ with $x_{\alpha_{2}}(rs)$
commutes with $x_{\alpha_{1}}(r),x_{\alpha_{3}}(s).$ For $G_{2}$, the long
roots form a subsystem of $A_{2}.$ For $B_{2},$ choose
$\\{\alpha_{1},\alpha_{4}\\}$ as a set of simple roots (see Figure 2). The
long root element $x_{\alpha_{3}}(2rs)$ is a commutator
$[x_{\alpha_{2}}(r),x_{\alpha_{4}}(s)]$ of the two short root elements, and
$x_{\alpha_{3}}(2rs)$ commutes with $x_{\alpha_{2}}(r),x_{\alpha_{4}}(s)$ (cf.
[37], Proposition of page 211). This shows that the arithmetic subgroup
$\Gamma$ contains a Heisenberg subgroup. The theorem is then proved by Lemma
8.3.

If we consider special length functions, general results can be proved. When
we consider the stable word lengths, the following is essentially already
known (cf. Polterovich [53], Corollary 1.1.D and its proof).

###### Proposition 8.4

Let $\Gamma$ be an irreducible non-uniform lattice in a semisimple connected,
Lie group without compact factors and with finite center of real rank $\geq
2.$ Assume that a group $G$ has a virtually poly-positive stable word length.
In other words, the group $G$ has a finite-index subgroup $H$ and a subnormal
series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=H$

such that every finitely generated subgroup of $H_{i}/H_{i+1}$
($i=0,1,\cdots,n-1$) has a purely positive stable word length. Then any group
homomorphism $f:\Gamma\rightarrow G$ has its image finite.

Proof. Without loss of generality, we assume that $f$ takes image in $H.$
Since a lattice is finitely generated, $\Gamma$ has its image in $H_{0}/H_{1}$
finitely generated. When the image has a purely positive word length, any
distorted element in $\Gamma$ must have trivial image in $H_{0}/H_{1}$ (see ).
Lubotzky, Mozes and Raghunathan [44] prove that irreducible non-uniform
lattices in higher rank Lie groups have non-trivial distortion elements (They
actually prove the stronger result that there are elements in the group whose
word length has logarithmic growth). Then a finite-index subgroup
$\Gamma_{0}<\Gamma$ will have image in $H_{1}$, since high-rank irreducible
lattices are quasi-simple. An induction argument finishes the proof.

When we consider the length given by quasi-cocyles, the following is also
essentially already known ( cf. Py [54], Prop. 2.2, following Burger-Monod
[18] [19]). Recall that a locally compact group has property (TT) if any
continuous rough action on a Hilbert space has bounded orbits (see [49], page
172). Burger-Monod proves that an irreducible lattice $\Gamma$ in a high-rank
semisimple Lie group has property (TT).

###### Proposition 8.5

Let $\Gamma$ be an irreducible lattice in a semisimple connected, Lie group
without compact factors and with finite center of real rank $\geq 2.$. Assume
that a group $G$ has a virtually poly-positive average norm for quasi-
cocycles. In other words, the group $G$ has a finite-index subgroup $H$ and a
subnormal series

$1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft
H_{0}=H$

such that every finitely generated subgroup of $H_{i}/H_{i+1}$
($i=0,1,\cdots,n-1$) has a purely positive length given by a quasi-cocycle
with values in Hilbert spaces. Then any group homomorphism
$f:\Gamma\rightarrow G$ has its image finite.

Proof. Note that a group $\Gamma$ has property (TT) if and only if
$H^{1}(\Gamma;E)=0$ and

$\ker(H_{b}^{2}(\Gamma;E)\rightarrow H^{2}(\Gamma;E))=0$

for any linear isometric action of $\Gamma$ on a Hilbert space $E.$ Here
$H_{b}^{2}(\Gamma;E)$ is the second bounded cohomology group. Suppose that
$u:\Gamma\rightarrow E$ is a quasi-cocyle. There is a bounded map
$v:\Gamma\rightarrow E$ and a $1$-cocycle $w:\Gamma\rightarrow E$ such that

$u=v+w,$

by Proposition 2.1 of Py [54]. Since $\Gamma$ has property T, there exists
$x_{0}\in E$ such that $w(\gamma)=\gamma x_{0}-x_{0}.$ Therefore, we have

$\displaystyle\frac{\|u(\gamma^{n})\|}{n}$ $\displaystyle=$
$\displaystyle\frac{\|v(\gamma^{n})+w(\gamma^{n})\|}{n}$ $\displaystyle=$
$\displaystyle\frac{\|v(\gamma^{n})+\gamma^{n}x_{0}-x_{0}\|}{n}$
$\displaystyle\leq$
$\displaystyle\frac{\|v(\gamma^{n})\|+2\|x_{0}\|}{n}\rightarrow 0.$

Without loss of generality, we assume that $G=H.$ Suppose that any finitely
generated subgroup of $H/H_{1}$ has a purely positive average norm $l$ given
by a cocycle. The composite

$\Gamma\overset{f}{\rightarrow}H\rightarrow H/H_{1}$

has a finite-index kernel $\Gamma_{0}$, since $l$ vanishes on infinite-order
elements of the image. This implies that $f(\Gamma_{0})$ lies in $H_{1}.$ A
similar argument proves that $\ker f$ is of finite index in the general case.

##  9 Rigidity of group homomorphisms on matrix groups

###  9.1 Steinberg groups over finite rings

Recall that a ring $R$ is right Artinian if any non-empty family of right
ideals contains minimal elements. A ring $R$ is semi-local if
$R/\mathrm{rad}(R)$ is right Artinian (see Bass’ K-theory book [3] page 79 and
page 86), where $\mathrm{rad}(R)$ is the Jacobson radical. Let $n$ be a
positive integer and $R^{n}$ the free $R$-module of rank $n$ with standard
basis. A vector $(a_{1},\ldots,a_{n})$ in $R^{n}$ is called _right unimodular_
if there are elements $b_{1},\ldots,b_{n}\in R$ such that
$a_{1}b_{1}+\cdots+a_{n}b_{n}=1$. The _stable range condition_
$\mathrm{sr}_{m}$ says that if $(a_{1},\ldots,a_{m+1})$ is a right unimodular
vector then there exist elements $b_{1},\ldots,b_{m}\in R$ such that
$(a_{1}+a_{m+1}b_{1},\ldots,a_{m}+a_{m+1}b_{m})$ is right unimodular. It
follows easily that $\mathrm{sr}_{m}\Rightarrow\mathrm{sr}_{n}$ for any $n\geq
m$. A semi-local ring has the stable range $\mathrm{sr}_{2}$ ( [3], page 267,
the proof of Theorem 9.1). A finite ring $R$ is right Artinian and thus has
$\mathrm{sr}_{2}.$ The stable range

$\mathrm{sr}(R)=\min\\{m:R\text{ has }\mathrm{sr}_{m+1}\\}.$

Thus $\mathrm{sr}(R)=1$ for a finite ring $R$.

We briefly recall the definitions of the elementary subgroups $E_{n}(R)$ of
the general linear group $\mathrm{GL}_{n}(R)$, and the Steinberg groups
$\mathrm{St}_{n}(R)$. Let $R$ be an associative ring with identity and $n\geq
2$ be an integer. The general linear group $\mathrm{GL}_{n}(R)$ is the group
of all $n\times n$ invertible matrices with entries in $R$. For an element
$r\in R$ and any integers $i,j$ such that $1\leq i\neq j\leq n,$ denote by
$e_{ij}(r)$ the elementary $n\times n$ matrix with $1s$ in the diagonal
positions and $r$ in the $(i,j)$-th position and zeros elsewhere. The group
$E_{n}(R)$ is generated by all such $e_{ij}(r),$ i.e.

$E_{n}(R)=\langle e_{ij}(r)|1\leq i\neq j\leq n,r\in R\rangle.$

Denote by $I_{n}$ the identity matrix and by $[a,b]$ the commutator
$aba^{-1}b^{-1}.$

The following lemma displays the commutator formulas for $E_{n}(R)$ (cf. Lemma
9.4 in [47]).

###### Lemma 9.1

Let $R$ be a ring and $r,s\in R.$ Then for distinct integers $i,j,k,l$ with
$1\leq i,j,k,l\leq n,$ the following hold:

  1. (1)

$e_{ij}(r+s)=e_{ij}(r)e_{ij}(s);$

  2. (2)

$[e_{ij}(r),e_{jk}(s)]=e_{ik}(rs);$

  3. (3)

$[e_{ij}(r),e_{kl}(s)]=I_{n}.$

By Lemma 9.1, the group $E_{n}(R)$ $(n\geq 3)$ is finitely generated when the
ring $R$ is finitely generated. Moreover, when $n\geq 3,$ the group $E_{n}(R)$
is normally generated by any elementary matrix $e_{ij}(1).$

The commutator formulas can be used to define Steinberg groups as follows. For
$n\geq 3,$ the Steinberg group $\mathrm{St}_{n}(R)$ is the group generated by
the symbols $\\{x_{ij}(r):1\leq i\neq j\leq n,r\in R\\}$ subject to the
following relations:

  1. (St$1$)

$x_{ij}(r+s)=x_{ij}(r)x_{ij}(s);$

  2. (St$2$)

$[x_{ij}(r),x_{jk}(s)]=x_{ik}(rs)$ for $i\neq k;$

  3. (St$3)$

$[x_{ij}(r),x_{kl}(s)]=1$ for $i\neq l,j\neq k.$

There is an obvious surjection $\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$
defined by $x_{ij}(r)\longmapsto e_{ij}(r).$

For any ideal $I\vartriangleleft R,$ let $p:R\rightarrow R/I$ be the quotient
map. Then the map $p$ induces a group homomorphism
$p_{\ast}:\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n}(R/I).$ Denote by
$\mathrm{St}_{n}(R,I)$ (resp., $E_{n}(R,I)$) the subgroup of
$\mathrm{St}_{n}(R)$ (resp., $E_{n}(R)$) normally generated by elements of the
form $x_{ij}(r)$ (resp., $e_{ij}(r)$) for $r\in I$ and $1\leq i\neq j\leq n.$
In fact, $\mathrm{St}_{n}(R,I)$ is the kernel of $p_{\ast}$ (cf. Lemma 13.18
in Magurn [47] and its proof). However, $E_{n}(R,I)$ may not be the kernel of
$E_{n}(R)\rightarrow E_{n}(R/I)$ induced by $p.$

###### Lemma 9.2

When $n\geq\mathrm{sr}(R)+2,$ the natural map
$\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n+1}(R)$ is injective. In
particular, when $R$ is finite, the Steinberg group $\mathrm{St}_{n}(R)$ is
finite for any $n\geq 3.$

Proof. Let $W(n,R)$ be the kernel of the natural map
$\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n+1}(R).$ When
$n\geq\mathrm{sr}(R)+2,$ the kernel $W(n,R)$ is trivial (cf. Kolster [41],
Theorem 3.1 and Cor. 2.10). When $n$ is sufficient large, the Steinberg group
$\mathrm{St}_{n}(R)$ is the universal central extension of $E_{n}(R)$ (cf.
[59], Proposition 5.5.1. page 240). Therefore, the kernel
$\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$ is the second homology group
$H_{2}(E_{n}(R);\mathbb{Z}).$ When $R$ is finite, both $E_{n}(R)$ and
$H_{2}(E_{n}(R);\mathbb{Z})$ are finite. Therefore, the group
$\mathrm{St}_{n}(R)$ is finite for any $n\geq 3.$

###  9.2 Rigidity of group homomorphisms on matrix groups

###### Theorem 9.3

Suppose that $G$ is a group satisfying that

1) $G$ has a purely positive length function, i.e. there is a length function
$l:G\rightarrow[0,\infty)$ such that $l(g)>0$ for any infinite-order element
$g;$ and

2) any torsion abelian subgroup of $G$ is finitely generated.

Let $R$ be an associative ring with identity and $\mathrm{St}_{n}(R)$ the
Steinberg group. Suppose that $S<\mathrm{St}_{n}(R)$ is a finite-index
subgroup. Then any group homomorphism $f:\mathrm{St}_{n}(R)\rightarrow G$ has
its image finite when $n\geq 3$.

Proof. Since any ring $R$ is a quotient of a free (non-commutative) ring
$\mathbb{Z}\langle X\rangle$ for some set $X$ and $\mathrm{St}_{n}(R)$ is
functorial with respect to the ring $R,$ we assume without loss of generality
that $R=$ $\mathbb{Z}\langle X\rangle$. We prove the case
$S=\mathrm{St}_{n}(R)\ $first. Let $x_{ij}=\langle x_{ij}(r):r\in R\rangle,$
which is isomorphic to the abelian group $R.$ Note that

$[x_{12}(1),x_{23}(1)]=x_{13}(1)$

and $x_{13}(1)$ commutes with $x_{12}(1)$ and $x_{23}(1).$ Lemma 5.2 implies
any length function vanishes on $x_{13}(1).$ By Lemma 1.4, the length
$l(f(x_{13}(1)))=0.$ Note that $x_{ij}(r)$ is conjugate to $x_{13}(r)$ for any
$r\in R$ and $i,j$ satisfying $1\leq i\neq j\leq n.$ Since $l$ is purely
postive, we get that $f(x_{12}(1))$ is of finite order. Let $I=\ker
f|_{x_{12}}.$ Then $I\neq\varnothing,$ as $f(x_{12}(1))$ is of finite order.
For any $x\in I,$ and $y\in R,$ we have

$x_{12}(xy)=[x_{13}(x),x_{32}(y)].$

Therefore,

$f(x_{12}(xy))=[f(x_{13}(x),f(x_{32}(y)))]=1$

and thus $xy\in I.$ Similarly, we have
$f(x_{12}(yx))=f([x_{13}(y),x_{32}(x)])=1.$ This proves that $I$ is a (two-
sided) ideal. Note that $f(e_{12})=R/I$ is a torsion abelian group. By the
assumption 2), the quotient ring $R/I$ is finite. Let $\mathrm{St}_{n}(R,I)$
be the normal subgroup of $\mathrm{St}_{n}(R)$ generated by $x_{ij}(r),r\in
I.$ There is a short exact sequence

$\begin{array}[]{ccc}1\rightarrow\mathrm{St}_{n}(R,I)\rightarrow&\mathrm{St}_{n}(R)\rightarrow&\mathrm{St}_{n}(R/I)\rightarrow
1.\end{array}$

Since $R/I$ is finite, we know that $\mathrm{St}_{n}(R/I)$ is finite by Lemma
9.2. This proves that $\mathop{\mathrm{I}m}f$ is finite since $f$ factors
through $\mathrm{St}_{n}(R/I)$. For general finite-index subgroup $S,$ we
assume $S$ is normal in $\mathrm{St}_{n}(R)$ after passing to a finite-index
subgroup of $S.$ A similar proof shows that $S$ contains
$\mathrm{St}_{n}(R,I)$ for some ideal $I$ with the quotient ring $R/I$ finite.
Therefore, the image $\mathop{\mathrm{I}m}f$ is finite.

###### Theorem 9.4

Suppose that $G$ is a group having a purely positive length function $l$. Let
$R$ be an associative ring of characteristic zero such that any nonzero ideal
is of a finite index (eg. the ring of algebraic integers in a number field).
Suppose that $S<\mathrm{St}_{n}(R)$ is a finite-index subgroup of the
Steinberg group. Then any group homomorphism $f:S\rightarrow G$ has its image
finite when $n\geq 3$.

Proof. The proof is similar to that of Theorem 9.3. Let $I=\ker f|_{x_{12}},$
where $x_{12}=S\cap\langle x_{12}(r):r\in R\rangle.$ Since $R$ is of
characteristic zero and the length $l(f(x_{12}(k)))=0$ for some integer $k,$
we have $f(x_{12}(k))$ is of finite order. Therefore,
$f(x_{12}(k^{\prime}))=1$ for some integer $k^{\prime},$ which proves that the
ideal $I$ is nozero. Since $I$ is of finite index in $R,$ we get that
$\mathrm{St}_{n}(R,I)$ is of finite index in $S.$ This finishes the proof.

Since the natural map $\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$ is surjective,
any group homomorphism $f:E_{n}(R)\rightarrow G$ can be lifted to be a group
homomorphism $\mathrm{St}_{n}(R)\rightarrow G.$ Moreover, a finite-index
subgroup $E$ of $E_{n}(R)$ is lifted to be a finite-index subgroup $S$ of
$\mathrm{St}_{n}(R).$ Theorem 0.4 and Theorem 0.6 follows Theorem 9.3 and
Theorem 9.4, by inductive arugments on the subnormal series as those of the
proofs of Theorem 0.3.

Proof of Corollary 0.5 and Corollary 0.7. For Corollary 0.5, it is enough to
check the two conditions for $G$ in Theorem 0.4. Lemma 3.6 proves that $G$ has
a purely positive length function. When $G$ is a $\mathrm{CAT}(0)$ group,
(i.e. $G$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ space), then
any solvable subgroup of $G$ is finitely generated (and actually virtually
abelian, see the Solvable Subgroup Theorem of [16], Theorem 7.8, page 249).
When $G$ is hyperbolic, it’s well-known that $G$ contains finitely many
conjugacy classes of finite subgroups and thus a torsion abelian subgroup is
finite (see [16], Theorem 3.2, page 459). Birman-Lubotzky-McCarthy [7] proves
that any abelian subgroup of the mapping class groups for orientable surfaces
is finitely generated. Bestvina-Handel [6] proves that every solvable subgroup
of $\mathrm{Out}(F_{k})$ has a finite index subgroup that is finitely
generated and free abelian. When $G$ is the diffeomorphism group
$\mathrm{Diff}(\Sigma,\omega),$ there is a subnormal series (see the proof of
Lemma 3.8)

$1\vartriangleleft\mathrm{Ham}(\Sigma,\omega)\vartriangleleft\mathrm{Diff}_{0}(\Sigma,\omega)\vartriangleleft\mathrm{Diff}(\Sigma,\omega),$

with subquoitents in $\mathrm{Ham}(\Sigma,\omega),$ $H_{1}(\Sigma,\mathbb{R})$
and the mapping class group $\mathrm{MCG}(\Sigma).$ Any abelian subgroup of a
finitely generated subgroup of these groups is finitely generated.

Corrolary 0.7 follows Theorem 9.4 and Lemma 3.6.

###### Remark 9.5

An infinite torsion abelian group may act properly on a simplicial tree (see
[16], Example 7.11, page 250). Therefore, condition 2) in Theorem 0.4 does not
hold for every group $G$ acting properly on a $\mathrm{CAT}(0)$ (or a Gromov
hyperbolic) space. We don’t know whether the condition 2) can be dropped.

##  10 Length functions on Cremona groups

Let $k$ be a field and $k(x_{1},x_{2},\cdots,x_{n})$ be the field of rational
functions in $n$ indeterminates over $k.$ It is well-known that the Cremona
group $\mathrm{Cr}_{n}(k)$ is isomorphic to the automorphism group
$\mathrm{Aut}_{k}(k(x_{1},x_{2},\cdots,x_{n}))$ of the field
$k(x_{1},x_{2},\cdots,x_{n})$.

###### Lemma 10.1

Let $f:k(x_{1},x_{2},\cdots,x_{n})\rightarrow k(x_{1},x_{2},\cdots,x_{n})$ be
given by $f(x_{1})=\alpha x_{1},f(x_{i})=x_{i}$ for some $0\neq\alpha\in k$
and any $i=2,\cdots,n.$ Then $f$ lies in the center of a Heisenberg subgroup.
In other words, there exists $g,h\in\mathrm{Cr}_{n}(k)$ such that
$[g,h]=ghg^{-1}h^{-1}=f,[g,f]=1$ and $[h,f]=1.$

Proof. Let $g,h:k(x_{1},x_{2},\cdots,x_{n})\rightarrow
k(x_{1},x_{2},\cdots,x_{n})$ be given by

$g(x_{1})=x_{1}x_{2},g(x_{i})=x_{i}(i=2,\cdots,n)$

and

$h(x_{1})=x_{1},h(x_{2})=\alpha^{-1}x_{2},h(x_{j})=x_{j}(j=3,\cdots,n).$

It can be directly checked that $[g,h]=f,[g,f]=1$ and $[h,f]=1.$

###### Lemma 10.2

Let $l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,\infty)$ be a length
function $(n\geq 2)$. Then $l$ vanishes on diagonal elements and unipotent
elements of $\mathrm{Aut}(\mathbb{P}_{k}^{n})=\mathrm{PGL}_{n+1}(k).$

Proof. Let $g=\mathrm{diag}(a_{0},a_{1},\cdots,a_{n})\in\mathrm{PGL}_{n+1}(k)$
be a diagonal element. Note that $l$ is subadditive on the diagonal subgroups.
In order to prove $l(g)=0,$ it is enough to prove that
$l(\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1))=0,$ where
$\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1)$ is the diagonal matrix with
$a_{i}$ in the $(i,i)$-th position and all other diagonal entries are $1.$ But
$\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1)$ is conjugate to
$\mathrm{diag}(1,\alpha,1,\cdots,1)$ for $\alpha=a_{i}.$ Lemma 10.1 implies
that $\mathrm{diag}(1,\alpha,1,\cdots,1)$ lies in the center of a Heisenberg
group. Therefore, $l(\mathrm{diag}(1,\alpha,1,\cdots,1))=0$ by Lemma 5.2. This
proves $l(g)=0.$ The vanishing of $l$ on unipotent elements follows Corollary
6.3 when the characteristic of $k$ is not $2$. When the characteristic of $k$
is $2,$ any unipotent element $A=I+u$ (where $u$ is nilpotent) is of finite
order. This means $l(A)=0$.

Proof of Theorem 0.8. When $k$ is algebraically closed, the Jordan normal form
implies that any element $g\in\mathrm{PGL}_{n}(k)$ is conjugate to the form
$sn$ with $s$ diagonal and $n$ the strictly upper triangular matrix. Moreover,
$sn=ns.$ Therefore, $l(f)\leq l(s)+l(n).$ By Lemma 10.2, $l(s)=l(n)=0$ and
thus $l(g)=0$.

Proof of Corollary 0.9. Let $f:\mathrm{Bir}(\mathbb{P}_{k}^{2})\rightarrow G$
be a group homomorphism. Suppose that $G$ has a purely positive length
function $l.$ By Theorem 0.8, the purely positive length function $l$ on $G$
will vanish on $f(\mathrm{PGL}_{3}(k)).$ Since $k$ is infinite and
$\mathrm{PGL}_{3}(k)$ is a simple group, we get that $\mathrm{PGL}_{3}(k)$
lies in the $\ker f.$ By Noether-Castelnuovo Theorem,
$\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is generated by $\mathrm{PGL}_{3}(k)$ and
an involution. Moreover, the $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is normally
generated by $\mathrm{PGL}_{3}(k).$ Therefore, the group homomorphism $f$ is
trivial. The general case is proved by an inductive argument on the subnormal
series of a finite-index subgroup of $G.$

###### Lemma 10.3

Let $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{n})$ $(n\geq 2)$ be the real
Cremona group. Any length function
$l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,\infty),$ which is continuous
on $\mathrm{PSO}(n+1)<\mathrm{Aut}(\mathbb{P}_{\mathbb{R}}^{n}),$ vanishes on
$\mathrm{PGL}_{n+1}(\mathbb{R}).$

Proof. By Lemma 10.2, the length function $l$ vanishes on diagonal matrices of
$\mathrm{PGL}_{n+1}(\mathbb{R}).$ Theorem 0.11 implies that $l$ vanishes on
the whole group $\mathrm{PGL}_{n+1}(\mathbb{R}).$

Acknowledgements

The author wants to thank many people for helpful discussions, including
Wenyuan Yang on a discussion of hyperbolic groups, C. Weibel on a discussion
on Steinberg groups of finite rings, Feng Su on a discussion of Lie groups,
Ying Zhang on a discussion of translation lengths of hyperbolic spaces, Enhui
Shi for a discussion on smooth measure-theoretic entropy.

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NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan
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