aa r X i v : . [ m a t h . G R ] J a n Length functions on groups and rigidity
Shengkui YeJanuary 25, 2021
Abstract
Let G be a group. A function l : G → [0 , ∞ ) is called a length function if(1) l ( g n ) = | n | l ( g ) for any g ∈ G and n ∈ Z ;(2) l ( hgh − ) = l ( g ) for any h, g ∈ G ; and(3) l ( ab ) ≤ l ( a ) + l ( b ) for commuting elements a, b. Such length functions exist in many branches of mathematics, mainly as stableword lengths, stable norms, smooth measure-theoretic entropy, translation lengthson CAT(0) spaces and Gromov δ -hyperbolic spaces, stable norms of quasi-cocycles,rotation numbers of circle homeomorphisms, dynamical degrees of birational mapsand so on. We study length functions on Lie groups, Gromov hyperbolic groups,arithmetic subgroups, matrix groups over rings and Cremona groups. As applica-tions, we prove that every group homomorphism from an arithmetic subgroup ofa simple algebraic Q -group of Q -rank at least 2 , or a finite-index subgroup of theelementary group E n ( R ) ( n ≥
3) over an associative ring, or the Cremona groupBir( P C ) to any group G having a purely positive length function must have its imagefinite. Here G can be outer automorphism group Out( F n ) of free groups, mappingclasses group MCG(Σ g ), CAT(0) groups or Gromov hyperbolic groups, or the groupDiff(Σ , ω ) of diffeomorphisms of a hyperbolic closed surface preserving an area form ω. The rigidity phenomena have been studied for many years. The famous Margulis super-rigidity implies any group homomorphism between irreducible lattices in semisimple Liegroups of real rank rk R ( G ) ≥ g ) has a finite image. Bridson and Wade [17] showedthat the same superrigidity remains true if the target is replaced with the outer automor-phism group Out( F n ) of the free group . Mimura [48] proves that every homomorphismfrom Chevalley group over commutative rings to MCG(Σ g ) or Out( F n ) has a finite image.Many other rigidity results can be found, e.g. [49] [18] [19] [33] [54] and [53]. In thisarticle, we study rigidity phenomena with the notion of length functions.Let G be a group. We call a function l : G → [0 , ∞ ) a length function if1) l ( g n ) = | n | l ( g ) for any g ∈ G and n ∈ Z ;2) l ( aga − ) = l ( g ) for any a, g ∈ G ;3) l ( ab ) ≤ l ( a ) + l ( b ) for commuting elements a, b. 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 necessar-ily finite) set S. For any g ∈ G, the word length φ S ( w ) = min { n | g = s s · · · s n , each s i ∈ S } is the minimal number of elements of S whose product is g. The stable lengthis defined as l ( g ) = lim n →∞ φ S ( g n ) n . • (Stable norms) Let M be a compact smooth manifold and G = Diff( M ) the dif-feomorphism group consisting of all self-diffeomorphisms. For any diffeomorphism f : M → M, let k f k = sup x ∈ M k D x f k , where D x f is the induced linear map between tangent spaces T x M → T f ( x ) M. Define l ( f ) = max { lim n → + ∞ log k f n k n , lim n → + ∞ log k f − n k n } . • (smooth measure-theoretic entropy) Let M be a C ∞ closed Riemannian manifoldand G = Diff µ ( M ) consisting of diffeomorphisms of M preserving a Borel probabilitymeasure µ. Let l ( f ) = h µ ( f ) be the measure-theoretic entropy, for any f ∈ G =Diff µ ( M ). • (Translation lengths) Let ( X, d ) be a metric space and G = Isom( X ) consisting ofisometries γ : X → X . Fix x ∈ X, define l ( γ ) = lim n →∞ d ( x, γ n x ) n . This contains the translation lengths on CAT(0) spaces and Gromov δ -hyperbolicspaces as special cases. • (average norm for quasi-cocycles) Let G be a group and E be a Hilbert space withan G -action by linear isometrical action. A function f : G → E is a quasi-cocyle ifthere exists C > k f ( gh ) − f ( g ) − gf ( h ) k < C for any g, h ∈ G. Let l : G → [0 , + ∞ ) be defined by l ( g ) = lim n →∞ k f ( g n ) k n . • (Rotation numbers of circle homeomorphisms) Let R be the real line and G =Home Z ( R ) = { f | f : R → R is a monotonically increasing homeomorphism suchthat f ( x + n ) = f ( x ) for any n ∈ Z } . For any f ∈ Home Z ( R ) and x ∈ [0 , , thetranslation number is defined as l ( f ) = lim n →∞ f n ( x ) − xn . (Asymptotic distortions) Let f be a C bv diffeomorphism of the closed interval [0 , S . (“bv” means derivative with finite total variation.) The asymptoticdistortion of f is defined (by Navas [50]) as l ( f ) = lim n →∞ var(log Df n ) . This gives a length function l on the group Diff bv ( M ) of C bv diffeomorphismsfor M = [0 ,
1] or S . • (Dynamical degree) Let C P n be the complex projective space and f : C P n C P n be a birational map given by( x : x : · · · : x n ) ( f : f : · · · : f n ) , where the f i ’s are homogeneous polynomials of the same degree without commonfactors. The degree of f is deg f = deg f i . Define l ( f ) = max { lim n →∞ log deg( f n ) n , lim n →∞ log deg( f − n ) n } . This gives a length function l : Bir( C P n ) → [0 , + ∞ ) . Here Bir( C P n ) is the groupof 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 thecondition 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 = Z ⋊ A Z be an abelian-by-cyclic group, where A ∈ SL ( Z ) .(i) When the absolute value of the trace | tr( A ) | > , any length function l : Z ⋊ A Z → R ≥ vanishes on Z . (ii) When | tr( A ) | = 2 and A = I , any length function l : Z ⋊ A Z → R ≥ vanishes onthe direct summand of Z spanned by eigenvectors of A . Corollary 0.2
Suppose that the semi-direct product G A = Z ⋊ A Z acts on a compactmanifold by Lipschitz homeomorphisms (or C -diffeomorphisms, resp.). The topologicalentropy h top ( g ) = 0 (or Lyapunov exponents of g are zero, resp.) for any g ∈ Z when | tr( A ) | > or any eigenvector g ∈ Z when | tr( A ) | = 2 . It is well-known that the central element in the integral Heisenberg group G A (for A = (cid:20) (cid:21) ) is distorted in the word metric. When the Heisenberg group G A acts on a C ∞ compact Riemannian manifold, Hu-Shi-Wang [36] proves that the topological entropyand all Lyapunov exponents of the central element are zero. These results are special casesof Theorem 0.1 and Corollary 0.2, by choosing special length functions.A length function l : G → [0 , ∞ ) is called purely positive if l ( g ) > g. A group G is called virtually poly-positive, if there is a finite-indexsubgroup H < G and a subnormal series1 = H n ⊳ H n − ⊳ · · · ⊳ H = H such that every finitely generated subgroup of each quotient H i /H i +1 ( i = 0 , ..., n − heorem 0.3 Let Γ be an arithmetic subgroup of a simple algebraic Q -group of Q -rankat least . Suppose that G is virtually poly-positive. Then any group homomorphism f : Γ → G has its image finite. Theorem 0.4
Let G be a group having a finite-index subgroup H < G and a subnormalseries H n ⊳ H n − ⊳ · · · ⊳ H = H satisfying that(i) every finitely generated subgroup of each quotient H i /H i +1 ( i = 0 , ..., n − has apurely positive length function, i.e. G is virtually poly-positive; and(ii) any torsion abelian subgroup in every finitely generated subgroup of each quotient H i /H i +1 ( i = 0 , ..., n − is finitely generated.Let R be a finitely generated associative ring with identity and E n ( R ) the elementarysubgroup. Suppose that Γ < E n ( R ) is finite-index subgroup. Then any group homomor-phism f : Γ → G has its image finite when n ≥ . Corollary 0.5
Let Γ be an arithmetic subgroup of a simple algebraic Q -group of Q -rankat least , or a finite-index subgroup of the elementary subgroup E n ( R ) ( n ≥ for anassociative ring R. Then any group homomorphism f : E → G has its image finite. Here G is one of the following groups: • a Gromov hyperbolic group, • CAT(0) group, • automorphism group Aut( F k ) of a free group , • outer automorphism group Out( F k ) of a free group or • mapping class group MCG(Σ g ) ( g ≥ . • the group Diff(Σ , ω ) of diffeomorphisms of a closed surface preserving an area form ω. Theorem 0.6
Suppose that G is virtually poly-positive. Let R be a finitely generatedassociative 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-indexsubgroup of the elementary group. Then any group homomorphism f : S → G has itsimage finite when n ≥ . Corollary 0.7
Let R be an associative ring of characteristic zero such that any nonzeroideal is of a finite index. Any group homomorphism f : E → G has its image finite, where E < E n ( R ) is finite-index subgroup and n ≥ . Here G is one of the followings: • Aut( F k ) , Out( F k ) , MCG(Σ g ) , • a hyperbolic group, • a CAT(0) group or more generally a semi-hyperbolic group, a group acting properly semi-simply on a CAT(0) space, or • a group acting properly semi-simply on a δ -hyperbolic space, • the group Diff(Σ , ω ) of diffeomorphisms of a hyperbolic closed surface preserving anarea form ω. Some relevent cases of Theorem 0.4 and Theorem 0.6 are already established in theliterature. Bridson and Wade [17] showed that any group homomorphism from an irre-ducible lattice in a semisimple Lie group of real rank ≥ g ) must have its image finite. However, Theorem 0.3 can never hold when Γ isa 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 tobe stable word lengths, Theorem 0.3 holds more generally for Γ non-uniform irreduciblelattices a semisimple Lie group of real rank ≥ G are given by a particular kindof quasi-cocyles, Theorem 0.3 holds more generally for Γ with property TT (cf. Py [54],Prop. 2.2). Haettel [33] prove that any action of a high-rank a higher rank lattice on aGromov-hyperbolic space is elementary (i.e. either elliptic or parabolic). Guirardel andHorbez [33] prove that every group homomorphism from a high-rank lattice to the outerautomorphism group of torsion-free hyperbolic group has finite image. Thom [57] (Corol-lary 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, ourtarget group G and the source group E n ( R ) (can be defined over any non-commutativering) in Theorem 0.4 are much more general. The inequalities of n in Theorem 0.4, Theo-rem 0.6 and Corollary 0.5, Corollary 0.7 can not be improved, since SL ( Z ) is hyperbolic.The group Γ in Corrolary 0.5 has Kazhdan’s property T (i.e. an arithmetic subgroupof a simple algebraic Q -group of Q -rank at least 2 , or a finite-index subgroup of the el-ementary subgroup E n ( R ) , n ≥ , for an associative ring R has Kazhdan’s property Tby [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 Γ withKazhdan’s property T. Franks and Handel [27] prove that any group homomorphism froma quasi-simple group containing a subgroup isomorphic to the three-dimensional integerHeisenberg group, to the group Diff(Σ , ω ) of diffeomorphisms of a closed surface preservingan area form ω, has its image finite (cf. Lemma 8.3).We now study length functions on the Cremona groups. Theorem 0.8
Let
Bir( P nk ) ( n ≥ be the group of birational maps on the projective space P nk over an algebraic closed field k . Any length function l : Bir( P nk ) → [0 , + ∞ ) vanisheson the automorphism group Aut( P nk ) = PGL n +1 ( k ) . When n = 2 , a result of Blanc and Furter [9] (page 7 and Proposition 4.8.10) impliesthat there are three length functions l , l , l on Bir( P k ) such that any element g ∈ Bir( P nk )satisfying l ( g ) = l ( g ) = l ( g ) = 0 is either finite or conjugate to an element in Aut( P k ) . This implies that the automorphism group Aut( P nk ) (when k = 2) is one of the ‘largest’subgroups of Bir( P nk ) on which every length function vanishes. Corollary 0.9
Let G be a virtually poly-positive group. Any group homomorphism f :Bir( P k ) → G is trivial, for an algebraic closed field k .
5n particular, Corrolary 0.9 implies that any quotient group of Bir( P k ) cannot actproperly semisimply neither on a Gromov δ -hyperbolic space nor a CAT(0) space. Thisis interesting, considering the following facts. There are (infinite-dimensional) hyperbolicspace and cubical complexes, on which Bir( P k ) acts isometrically (see [21], Section 3.1.2and [45]). The Cremona group Bir( P k ) is sub-quotient universal: every countable groupcan be embedded in a quotient group of Bir( P k ) (see [21], Theorem 4.7). Moreover, Blanc-Lamy-Zimmermann [10] (Theorem E) proves that when n ≥ , there is a surjection fromBir( P nk ) onto a free product of two-element groups Z / . This means that Corrolary 0.9can never hold for higher dimensional Cremona groups.As byproducts, we give characterizations of length functions on Lie groups. Our nextresult is that there is essentially only one length function on the special linear groupSL ( R ): Theorem 0.10
Let G = SL ( R ) . Any length function l : G → [0 , + ∞ ) continuous on thesubgroup SO (2) and the diagonal subgroup is proposional to the translation function τ ( g ) := inf x ∈ X d ( x, gx ) , where X = SL ( R ) / SO(2) is the upper-half plane.
More generally, we study length functions on Lie groups. Let G be a connectedsemisimple 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 thecentralizers C K ( A ). Our second result shows that a length function l on G is uniquelydetermined by its image on A. Theorem 0.11
Let G be a connected semisimple Lie group whose center is finite with anIwasawa decomposition G = KAN . Let W be the Weyl group.(i) Any length function l on G that is continuous on the maximal compact subgroup K is determined by its image on A. (ii) Conversely, any length function l on A that is W -invariant (i.e. l ( w · a ) = l ( a ) )can be extended to be a length function on G that vanishes on the maximal compactsubgroup K. The proofs of Theorems 0.10 0.11 are based the Jordan-Chevalley decompositions ofalgebraic groups and Lie groups. We will prove that any length function on a Heisenberggroup vanishes on the central elements (see Lemma 5.2). This is a key step for many otherproofs. Based this fact, we prove Theorems 0.3, 0.6, 0.4, 0.8 by looking for Heisenbergsubgroups. In Section 1, we give some elementary facts on the length functions. InSection 2, we discuss typical examples of length functions. In later sections, we studylength functions on Lie groups, algebraic groups, hyperbolic groups, matrix groups andthe Cremona groups. 6
Basic properties of length functions
Definition 1.1
Let G be a group. A function l : G → [0 , ∞ ) is called a length functionif 1) l ( g n ) = | n | l ( g ) for any g ∈ G and n ∈ Z . l ( aga − ) = l ( g ) for any a, g ∈ G. l ( ab ) ≤ l ( a ) + l ( b ) for commuting elements a, b. Lemma 1.2
Any torsion element g ∈ G has length l ( g ) = 0 . Proof.
Note that l (1) = 2 l (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 ∈ V for any v, w ∈ V and any non-negative real numbers a, b ≥ . Lemma 1.3
The set
Func( G ) of all length functions on a group G is a convex cone. Proof.
It is obvious that for two functions l , l on G, a non-negative linear combination al + bl is a new length function. Lemma 1.4
Let f : G → H be a group homomorphism between two groups G and H. For any length function l : H → [0 , ∞ ) , the composite l ◦ 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 Out( G ) = Aut( G ) / Inn( G ) be the outer automorphismgroup. Then Out( G ) acts on the set Func( G ) of all length functions by pre-compositions l l ◦ g, where l ∈ Func( G ) , g ∈ Out( G ) . This action preserves scalar multiplications and linearcombinations (with non-negative coefficients). Proof.
For an inner automorphism I g : G → G given by I g ( h ) = ghg − , the length func-tion l ◦ I g = l since l is invariant under conjugation. Therefore, the outer automorphismgroup Out( G ) has an action on Func( G ) . It is obvious that the pre-compositions preservescalar multiplications and linear combinations with non-negative coefficients.
Definition 1.6
A length function l : G → [0 , ∞ ) is primitive if it is not a composite l ′ ◦ f for a non-trivial surjective group homomorphism f : G ։ H and a length function l ′ : H → [0 , ∞ ) . Lemma 1.7
Suppose that a length function l : G → [0 , ∞ ) vanishes on a central subgroup H < G.
Then l factors through the quotient group G/H.
In other words, there exists alength function l ′ : G/H → [0 , ∞ ) such that l = l ′ ◦ q, where q : G → G/H is the quotientgroup homomorphism.
Proof.
Write G = ∪ gH, the union of left cosets. For any h ∈ H, we have l ( gh ) ≤ l ( g ) + l ( h ) = l ( g ) and l ( g ) = l ( ghh − ) ≤ l ( gh ) . Therefore, l ( gh ) = l ( g ) for any h ∈ H. Define l ′ ( gH ) = l ( g ) . Then l ′ is a length function on the quotient group G/H.
The requiredproperty follows the definition easily. 7 orollary 1.8
Suppose that a group G has non-trivial finite central subgroup Z ( G ) . Anylength 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 ⊳ G isof finite index. Then any non-vanishing length function l : G → [0 , ∞ ) is primitive. Proof.
Suppose that l is a composite l ′ ◦ f for a non-trivial surjective group homo-morphism f : G ։ H and a length function l ′ : H → [0 , ∞ ) . By the assumption of G, the quotient group H is finite. This implies that l ′ and thus l vanishes, which is acontradiction. Theorem 1.10
Let Γ be an irreducible lattice in a connected irreducible semisimple Liegroup of real rank ≥ . Then any non-vanishing length function l : Γ → [0 , ∞ ) factorsthrough a primitive function on Γ /Z (Γ) . Proof.
By the Margulis-Kazhdan theorem (see [62], Theorem 8.1.2), any normal subgroup N of Γ either lies in the center of Γ (and hence it is finite) or the quotient group Γ /N isfinite. Corollary 1.8 implies that l factors through a length function l ′ on Γ /Z (Γ) . Theprevious lemma 1.9 implies that l ′ is primitive. Let’s see a general example first. Let G be a goup and f : G → [0 , + ∞ ) be a functionsatisfying f ( gh ) ≤ f ( g ) + f ( h ) and f ( g ) = f ( g − ) for any elements g, h ∈ G. Define l : G → [0 , + ∞ ) by l ( g ) = lim n →∞ f ( g n ) n for any g ∈ G. Lemma 2.1
The function l is a length function in the sense of Definition 1.1. Proof.
For any g ∈ G, and natural numbers n, m, we have f ( g n + m ) ≤ f ( g n ) + f ( g m ) . Thismeans that { f ( g n ) } ∞ n =1 is a subadditive sequence and thus the limit lim n →∞ 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 ∈ G. We have l ( hgh − ) = lim n →∞ f ( hg n h − ) n ≤ lim n →∞ f ( h ) + f ( g n ) + f ( h − ) n = lim n →∞ f ( g n ) n = l ( g ) . Similarly, we have l ( g ) = l ( h − ( hgh − ) h ) ≤ l ( hgh − ) and thus l ( g ) = l ( hgh − ) . Forcommuting elements a, b, we have ( ab ) n = a n b n . Therefore, l ( ab ) = lim n →∞ f (( ab ) n ) n = lim n →∞ f ( a n b n ) n ≤ lim n →∞ f ( a n ) + f ( b n ) n ≤ l ( a ) + l ( b ) . Many (but not all) length functions l come from subadditive functions f. .1 Stable word lengths Let G be a group generated by a (not necessarily finite) set S satisfying s − ∈ S foreach s ∈ S. For any g ∈ G, the word length φ S ( w ) = min { n | g = s s · · · s n , each s i ∈ S } is the minimal number of elements of S whose product is g. The stable length l ( g ) = lim n →∞ φ S ( g n ) n . Since φ S ( g n ) is subadditive, the limit always exists. Lemma 2.2
The stable length l : G → [0 , + ∞ ) is a length function in the sense ofDefinition 1.1. Proof.
From the definition of the word length φ S , it is clear that φ S ( gh ) ≤ φ S ( g ) + φ S ( h )and φ S ( g ) = φ S ( g − ) for any g, h ∈ 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]). Let G be a group generated by a finite set S satisfying s − ∈ S for each s ∈ S. Sup-pose | · | S is the word length of ( G, S ) . For any automorphism α : G → G, define l ′ ( α ) = max {| α ( s i ) | S : s i ∈ S } . Let l ( α ) = lim n →∞ log l ′ ( α n ) n . This number l ( α ) is called thealgebraic entropy of α (cf. [40], Definition 3.1.9, page 114). Lemma 2.3
Let
Aut( G ) be the group of automorphisms of G. The function l : Aut( G ) → [0 , + ∞ ) is a length function in the sense of Definition 1.1. Proof.
Since α ( s i ) − = α − ( s i ) for any s i ∈ S, we know that l ′ ( α ) = l ′ ( α − ) . Foranother automorphism β : G → G, let l ′ ( β ) = | β ( s i ) | S for some s i ∈ S. Suppose that β ( s i ) = s i s i · · · s i k with k = l ′ ( β ) . Then | ( αβ )( s i ) | S = | α ( s i ) α ( s i ) · · · α ( s i k ) | S ≤ l ′ ( α ) k. This proves that l ′ ( αβ ) ≤ l ′ ( α ) l ′ ( β ) . The claim is proved by Lemma 2.1.Fix g ∈ G. For any automorphism α : G → G, define b n = | α n ( g ) | S . Suppose that g = s s · · · s k with k = | g | S . Note that b n = | α n ( g ) | S = | α n ( s ) α n ( s ) · · · α n ( s k ) | S ≤ l ′ ( α n ) | g | S . Therefore, we have lim sup n →∞ log b n n ≤ l ( α ) . This implies that l ( α ) is an upper bounded for growth rate of {| α n ( g ) | S } . The growthrate is studied a lot in geometric group theory (for example, see [43] for growth of auto-morphisms of free groups).
For a square matrix A, the matrix norm k A k = sup k x k =1 k Ax k . Define the stable norm s ( A ) = lim n → + ∞ log k A n k n . Since k AB k ≤ k A kk B k for any two matrices A, B, the sequence { log k A n k} ∞ n =1 is subadditive and thus the limit exists. Lemma 2.4
Let G = GL n ( R ) be the general linear group. The function l : G → [0 , + ∞ ) defined by l ( g ) = max { s ( g ) , s ( g − ) } is a length function in the sense of Definition 1.1. roof. From the definition of the matrix norm , it is clear that log k gh k ≤ log k g k +log k h k for any g, h ∈ G. Then l ( g ) = max { s ( g ) , s ( g − ) } is a length function by Lemma 2.1.Let M be a compact smooth manifold and Diff( M ) the diffeomorphism group consist-ing of all self-diffeomorphisms. For any diffeomorphism f : M → M, let k f k = sup x ∈ M k D x f k , where D x f is the induced linear map between tangent spaces T x M → T f ( x ) M. Define l ( f ) = max { lim n → + ∞ log k f n k n , lim n → + ∞ log k f − n k 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. Thefunction l : G → [0 , + ∞ ) is a length function in the sense of Definition 1.1. For an f -invariant Borel probability measure µ on M, it is well known (see [O]) thatthere exists a measurable subset Γ f ⊂ M with µ (Γ f ) = 1 such that for all x ∈ Γ f and u ∈ T x M, the limit χ ( x, u, f ) = lim 1 n log k D x f n ( u ) k exists and is called Lyapunov exponent of u at x. From the definitions, we know that χ ( x, u, f ) ≤ l ( f ) for any x ∈ Γ f and u ∈ T x M. Let T : X → X be a measure-preserving map of the probability space ( X, B , m ) . For afinite-sub- σ -algebra A = { A , A , ..., A k } of B , denote by H ( A ) = − X m ( A i ) log m ( A i ) ,h ( T, A ) = lim 1 n H ( ∨ n − i =0 T − i A ) , where ∨ n − i =0 T − i A is a set consisting of sets of the form ∩ n − i =0 T − i A j i . The entropy of T isdefined as h m ( T ) = sup h ( T, A ) , where the supremum is taken over all finite sub-algebra A of B . For more details, see Walters [58] (Section 4.4).
Lemma 2.6
Let M be a C ∞ closed Riemannian manifold and G = Diff µ ( M ) consistingof diffeomorphisms of M preserving a Borel probability measure µ. The entropy h µ is alength function on Diff µ ( M ) in the sense of Definition 1.1. Proof.
For any f, g ∈ Diff µ ( M ) and integer n, it is well-known that h µ ( f n ) = | n | h µ ( f )and h µ ( f ) = h µ ( gf g − ) (cf. [58], Theorem 4.11 and Theorem 4.13). Hu [35] proves that h µ ( f g ) ≤ h µ ( f ) + h µ ( g ) when f g = gf. .5 Stable translation length on metric spaces Let (
X, d ) be a metric space and γ : X → X an isometry. Fix x ∈ X. Note that d ( x, γ γ x ) ≤ d ( x, γ x ) + d ( γ x, γ γ x ) = d ( x, γ x ) + d ( x, γ x ) and d ( x, γ x ) = d ( x, γ − x )for any isometries γ , γ . Define l ( γ ) = lim n →∞ d ( x, γ n x ) n . For any y ∈ X, we have d ( x, γ n x ) ≤ d ( x, y ) + d ( y, γ n y ) + d ( γ n y, γ n x )= 2 d ( x, y ) + d ( y, γ n y )and thus lim n →∞ d ( x,γ n x ) n ≤ lim n →∞ d ( y,γ n y ) n . Similarly, we have the other directionlim n →∞ d ( y, γ n y ) n ≤ lim n →∞ d ( x, γ n x ) n . This shows that the definition of l ( γ ) 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 → [0 , + ∞ ) defined by g l ( g ) as above is a length function in the sense ofDefinition 1.1. Proof.
This follows Lemma 2.1.
In this subsection, we will prove that the translation length on a CAT(0) space defines alength function. First, let us introduce some notations. Let (
X, d X ) be a geodesic metricspace, i.e. any two points x, y ∈ X can be connected by a path [ x, y ] of length d X ( x, y ).For three points x, y, z ∈ X, the geodesic triangle ∆( x, y, z ) consists of the three vertices x, y, z and the three geodesics [ x, y ] , [ y, z ] and [ z, x ] . Let R be the Euclidean plane withthe standard distance d R and ¯∆ a triangle in R with the same edge lengths as ∆. Denoteby ϕ : ∆ → ¯∆ the map sending each edge of ∆ to the corresponding edge of ¯∆ . The space X is called a CAT(0) space if for any triangle ∆ and two elements a, b ∈ ∆ , we have theinequality d X ( a, b ) ≤ d R ( ϕ ( a ) , ϕ ( b )) . The typical examples of CAT(0) spaces include simplicial trees, hyperbolic spaces, prod-ucts 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 ∈ Isom( X ), letMinset( g ) = { x ∈ X : d ( x, gx ) ≤ d ( y, gy ) for any y ∈ X } and let τ ( g ) = inf x ∈ X d ( x, gx ) be the translation length of g. When the fixed-point setFix( g ) = ∅ , we call g elliptic. When Minset( g ) = ∅ and d X ( x, gx ) = τ ( g ) > x ∈ Minset( g ) , we call g hyperbolic. The group element g is called semisimple if theminimal set 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 ∈ C can connected by the geodesicsegment [ x, y ] ⊂ C. A group G is called CAT(0) if G acts properly discontinuously and11ocompactly on a CAT(0) space X . In such a case, any infinite-order element in G actshyperbolically on X. For more details on CAT(0) spaces, see the book of Bridson andHaefliger [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. Forcompleteness, we give details here.
Lemma 2.8
Let γ : X → X be an isometry of a complete CAT(0) space X. For any x ∈ X, we have τ ( γ ) := inf x ∈ X d ( γx, x ) = lim k →∞ d ( γ k x , x ) k . Proof.
For any p = x ∈ X, let m be the middle point of [ p, γp ] . We have that d ( m, γm ) ≤ d ( p, γ p ) by the convexity of length functions. Therefore, d ( p, γ p ) ≥ τ ( γ ) and τ ( γ ) ≥ τ ( γ ) . Note that d ( p, γ p ) ≤ d ( p, γp ) + d ( γp, γ p ) = 2 d ( p, γp ) and thus τ ( γ ) ≤ τ ( γ ) . Inductively, we have 2 n τ ( γ ) ≤ d ( p, γ n p ) ≤ n d ( p, γp ) . Note that the limit lim k →∞ d ( γ k p,p ) k exists and is independent of p (see the previous sub-section). Therefore, the limit lim k →∞ d ( γ k p,p ) k equals to τ ( γ ) . Corollary 2.9
Let X be a complete CAT(0) space and G a group acting on X by isome-tries. For any g ∈ G, define τ ( g ) = inf x ∈ X d ( x, gx ) as the translation length. Then τ : G → [0 , + ∞ ) is a length function in the sense of Definition 1.1. Proof.
This follows Lemma 2.8 and Lemma 2.7. δ -hyperbolic spaces Let δ > . A geodesic metric space X is called Gromov δ -hyperbolic if for any geodesictriangle ∆ xyz one side [ x, y ] is contained a δ -neighborhood of the other two edges [ x, z ] ∪ [ y, z ] . Fix x ∈ X. Any isometry γ : X → X is called elliptic if { γ n x } ne Z is bounded. Ifthe orbit map Z → X given by n γ n x is quasi-isometric (i.e. there exists A ≥ B ≥ A | n − m | − B ≤ d X ( γ n x , γ m x ) ≤ A | n − m | + B for any integers n, m ), we call γ is hyperbolic. Otherwise, we call γ is parabolic. Define l ( γ ) = lim n →∞ d ( γ n x ,x ) n . For any group G acts isometrically on a δ -hyperbolic space, thefunction l : G → [0 , ∞ ) is a length function by Lemma 2.7. A finitely generated group G is Gromov δ -hyperbolic if for some finite generating set S, the Caley graph Γ( G, S ) isGromov δ -hyperbolic. Any infinite-order element g in a Gromov δ -hyperbolic group ishyperbolic and thus has positive length l ( g ) > .8 Quasi-cocycles Let G be a group and ( E, k · k ) be a normed vector space with an G -action by linearisometries. A function f : G → E is a quasi-cocyle if there exists C > k f ( gh ) − f ( g ) − gf ( h ) k < C for any g, h ∈ G. Let l : G → [0 , + ∞ ) be defined by l ( g ) = lim n →∞ k f ( g n ) k n . Note that k f ( g n + m ) k ≤ k f ( g n ) k + k f ( g m ) k + C for any integers n, m ≥ . This generalsubadditive property implies that the limit lim n →∞ k f ( g n ) k n exists (see [56], Theorem 1.9.2,page 22). We call l the average norm. Many applications of quasi-cocycles can be foundin [49]. Lemma 2.10
For any quasi-cocyle f : G → E, the average norm l is a length function. Proof.
For any natural number n, we have k f (1) − f ( g − n ) − g − n f ( g n ) k < C and thus k f (1) − f ( g − n ) − g − n f ( g n ) n k < Cn . Taking the limit, we have lim n →∞ k f ( g − n ) k n = lim n →∞ k f ( g n ) k n . Therefore, for any k ∈ Z , we have l ( g k ) = lim n →∞ k f ( g kn ) k n = | k | l ( g ) . For any h ∈ G, wehave k f ( hg n h − ) k ≤ k f ( h ) k + k f ( h − ) k + k f ( g n ) k + 2 C. Therefore, we have l ( hgh − ) = lim n →∞ k f ( hg n h − ) k n ≤ lim n →∞ k f ( g n ) k n = l ( g ) . Similarly, we have l ( g ) = l ( h − ( hgh − ) h ) ≤ l ( hgh − ) . When g, h commutes, we have l ( gh ) = lim n →∞ k f (( gh ) n ) k n = lim n →∞ k f ( g n h n ) k n ≤ lim n →∞ k f ( g n ) k + k f ( h n ) k + Cn = l ( g ) + l ( h ) . Let R be the real line and Home Z ( R ) = { f | f : R → R is a monotonically increasinghomeomorphism such that f ( x + n ) = f ( x ) for any n ∈ Z } . For any f ∈ Home Z ( R ) and x ∈ [0 , , the translation number is defined as l ( f ) = lim n →∞ f n ( x ) − xn . It is well-known that l ( f ) exists and is independent of x (see [51], Prop. 2.22, p.31). Notethat every f ∈ Home Z ( R ) induces an orientation-preserving homeomorphism of the circle S . roposition 2.11 The absolute value of the translation number | l | : Home Z ( R ) → [0 , ∞ ) is a length function in the sense of Definition 1.1. Proof.
For any f ∈ Home Z ( R ) and k ∈ Z \{ } , we have that l ( f k ) = lim n →∞ f kn ( x ) − xn = k lim n →∞ f kn ( x ) − xnk = kl ( f ) . For any a ∈ Home Z ( R ) , we have that | l ( af a − ) − l ( f ) | = lim n →∞ | af n ( a − x ) − x − f n ( x ) + xn | = lim n →∞ | af n ( a − x ) − f n ( a − x ) + f n ( a − x ) − f n ( x ) n | = 0 , since a is bounded on [0 ,
1] and | f n ( a − x ) − f n ( x ) |≤ | a − x − x | . For commuting elements f, g ∈ Home Z ( R ) , we have that l ( f g ) = lim n →∞ f n ( g n ( x )) − xn = lim n →∞ f n ( g n ( x )) − g n ( x ) + g n ( x ) − xn . Suppose that g n ( x ) = k n + x n for k n ∈ Z and x n ∈ [0 , . Thenlim n →∞ f n ( g n x ) − g n ( x ) + g n ( x ) − xn = lim n →∞ f n (0) − g n ( x ) − xn = l ( f ) + l ( g ) . Therefore, we get | l ( f g ) | ≤ | l ( f ) | + | l ( g ) | . Remark 2.12
It is actually true that the rotation number l is multiplicative on anyamenable group (see [51], Prop. 2.2.11 and the proof of Prop. 2.2.10, page 36). Thisimplies that the absolute rotation number | l | is subadditive on any amenable group. Inother words, for any amenable group G <
Home Z ( R ) and any g, h ∈ G we have | l ( gh ) | ≤| l ( g ) | + | l ( h ) | . Let f be a C bv diffeomorphism of the closed interval [0 ,
1] or the circle S . (“bv” meansderivative with finite total variation.) The asymptotic distortion of f is defined as l ( f ) = dist ∞ ( f ) = lim n →∞ var(log Df n ) . It’s proved by Eynard-Bontemps amd Navas ([24], pages 7-8) that(1) dist ∞ ( f n ) = | n | dist ∞ ( f ) for all n ∈ Z ;(2) dist ∞ ( hf h − ) = dist ∞ ( f ) for every C bv diffeomorphism h ;(3) dist ∞ ( f ◦ g ) ≤ dist ∞ ( f ) + dist ∞ ( g ) for commuting f, g. Therefore, the asymptotic distortion is a length function l on the group Diff bv ( M )of C bv diffeomorphisms for M = [0 ,
1] or S . .11 Dynamical degrees of Cremona groups Let k be a field and P nk = k n +1 \{ } / { λ ∼ λx : λ = 0 } be the projective space. A rationalmap from P nk to itself is a map of the following type( x : x : · · · : x n ) ( f : f : · · · : f n )where the f i ’s are homogeneous polynomials of the same degree without common fac-tor. The degree of f is deg f = deg f i . A birational map from P nk to itself is a ra-tional map f : P nk P nk such that there exists a rational map g : P nk P nk suchthat f ◦ g = g ◦ f = id . The group Bir( P nk ) of birational maps is called the Cremonagroup (also denoted as Cr n ( k )). It is well-known that Bir( P nk ) is isomorphic to the groupAut k ( k ( x , x , · · · , x n )) of self-isomorphisms of the field k ( x , x , · · · , x n ) of the rationalfunctions in n indeterminates over k. The (first) dynamical degree λ ( f ) of f ∈ Bir( P nk ) isdefined as λ ( f ) = max { lim n →∞ deg( f n ) n , lim n →∞ deg( f − n ) n } . Since deg( f n ) n is sub-multiplicative, the limit exists. Lemma 2.13
Let l ( f ) = log λ ( f ) . Then l : Bir( P nk ) → [0 , + ∞ ) is a length function. Proof.
Without loss of generality, we assume that λ ( f ) = lim n →∞ deg( f n ) n , whilethe other case can be considered similarly. For any k ∈ N , it is easy that l ( f k ) =lim n →∞ log deg f nk n = kl ( f ) . For any h ∈ Bir( P nk ) , we have l ( hf h − ) = lim n →∞ log deg hf n h − n = lim n →∞ log deg f nk n = l ( f ) . For commuting maps f, g, we have ( f g ) n = f n g n . Therefore, l ( f g ) = lim n →∞ log deg f n g n n ≤ lim n →∞ log deg f n n + lim n →∞ 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 lengthfunction l ( f ) is given by the translation length τ ( f ) on an (infinite-dimensional) Gromov δ -hyperbolic space (see Blanc-Cantat [8], Theorem 4.4). Some other length functions arestudied by Blanc and Furter [9] for groups of birational maps, eg. dynamical number ofbase-points and dynamical length. Definition 3.1
A length function l on a group G is said to be purely positive if l ( g ) > for any infinite-order element g ∈ G. In this section, we show that the (Gromov) hyperbolic group, mapping class group andouter automorphism groups of free groups have purely positive length functions. First,let us recall the relevant definitions.A geodesic metric space X is δ -hyperbolic (for some real number δ >
0) if for anygeodesic triangle ∆ xyz in X, one side is contained the δ -neighborhood of the other twosides. A group G is (Gromov) hyperbolic if G acts properly discontinuously and cocom-pactly on a δ -hyperbolic space X . 15 efinition 3.2 (i) An element g in a group G is called primitive if it cannot be writenas a proper power α n , where α ∈ G and | n | ≥ (ii) A group G has unique-root property if every infinite-order element g is a properpower of a unique (up to sign) primitive element, i.e. g = γ n = γ m for primitiveelements γ, γ will imply γ = γ ± . 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 = g ∈ G. Suppose that g = γ n = γ m for primitive elements γ and γ . 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 isfree. Since G is torsion-free, the group C G ( g ) is thus free and thus cyclic, say generatedby t . Since γ and γ ′ are primitive, they are t ± . Remark 3.4
The previous lemma does not hold for general hyperbolic groups with tor-sions. For example, let G = Z / × Z . We have (0 ,
2) = (0 , = (1 , and (0 , , (1 , are both primitive. For a group G, let P ( G ) be the set of all primitive elements. We call two primitiveelements γ, γ ′ are general conjugate if there exists g ∈ G such that gγg − = γ ′ or gγ − g = γ ′ . Let CP( G ) be the general conjugacy classes of primitive elements. For a set S, let S R be the set of all real functions on S . The convex polyhedral cone spanned by S is thesubset { P s ∈ S a s s | a s ≥ } ⊂ S 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 CP( G ) . Proof.
Let l be a length function on G. Then l gives an element P s ∈ S a s s in the convexpolyhedral cone by a s = l ( s ) . Conversely, for any general conjugacy classes [ s ] ∈ CP( G )with s a primitive element, let l s be the function defined by l s ( s ± ) = 1 and l s ( γ ) = 0 forelement γ in any other general conjugacy classes. For any 1 = g ∈ G, there is a unique(up to sign) primitive element γ such that g = γ n . Define l s ( g ) = | n | l s ( γ ) . Then l s satisfiesconditions (1) and (2) in Definition 1.1. The condition (3) is satisfied automatically, sinceany commuting pair of elements a, b generate a cyclic group in a torsion-free hyperbolicgroup. Any element P s ∈ 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 Aut( F k ) of a free group ; • outer automorphism group Out( F k ) of a free group or • mapping class group MCG(Σ g,m ) (where Σ g,m is an oriented surface of genus g and m punctures); • a hyperbolic group, • a CAT(0) group or more generally a semi-hyperbolic group, • a group acting properly semi-simply on a CAT(0) space, • a group acting properly semi-simply on a δ -hyperbolic space.Then G has a purely positive length function. Proof.
Note that hyperbolic groups and CAT(0) groups are semihyperbolic (see [16],Prop. 4.6 and Cor. 4.8, Chapter III.Γ). For a semihyperbolic group G acting a metricspace X (actually X = G ), the translation τ is a length function by Lemma 2.7. Moreover,for any infinite-order element g ∈ G, the length τ ( g ) > δ -hyperbolic space),the translation l ( γ ) = lim n →∞ d ( x, γ n x ) n is a length function (cf. Lemma 2.7). For any hyperbolic γ , we get l ( γ ) > . For anyelliptic γ, it is finite-order since the action is proper.Alibegovic [1] proves that the stable word length of Aut( F n ) , Out( F n ) are purely pos-itive. Farb, Lubotzky and Minsky [25] prove that Dehn twists and more generally allelements of infinite order in MCG(Σ g,m ) have positive translation length. Definition 3.7
A group G is called poly-positive (or has a poly-positive length), if thereis a subnormal series H n ⊳ H n − ⊳ · · · ⊳ H = G such that every finitely generated subgroup of the quotient H i /H i +1 ( i = 0 , ..., n − hasa purely positive length function. Recall that a group G is poly-free, if there is a subnormal series 1 = H n ⊳ H n − ⊳ · · · ⊳ H = G such that the successive quotient H i /H i +1 is free ( i = 0 , ..., n − . Sincea free group is hyperbolic, it has a purely positive length function. This implies that apoly-free group is poly-positive. A group is said to have a virtual property if there is afinite-index subgroup has the property.Let Σ be a closed oriented surfaec endowed with an area form ω. Denote by Diff(Σ , ω )the group of diffeomorphisms preserving ω and Diff (Σ , ω ) the subgroup consisting ofdiffeomorphisms isotopic to the identity. Lemma 3.8
When the genuse of Σ is greater than , the group Diff (Σ , ω ) and Diff(Σ , ω ) is poly-positive. Proof.
This is eseentially proved by Py [54] (Section 1). There is a group homomorphism α : Diff (Σ , ω ) → H (Σ , R )with ker α = Ham(Σ , ω ) the group of Hamiltonian diffeomorphisms of Σ . Polterovich [53](1.6.C.) proves that any finitely generated group of Ham(Σ , ω ) has a purely positive stableword length. Since the quotient group Diff(Σ , ω ) / Diff (Σ , ω ) is a subgroup of the mappingclass group MCG(Σ) , which has a purely positive stable word length by Farb-Lubotzky-Minsky [25], the group Diff(Σ , ω ) is poly-positive.17 Vanishing of length functions on abelian-by-cyclicgroups
We will need the following result proved in [28].
Lemma 4.1
Given a group G , let l : G → [0 , + ∞ ) be function such that1) l ( e ) = 0; l ( x n ) = | n | l ( x ) for any x ∈ G, any n ∈ Z ; l ( xy ) ≤ l ( x ) + l ( y ) for any x, y ∈ G. Then there exist a real Banach space ( B , kk ) and a group homomorphism ϕ : G → B such that l ( x ) = k ϕ ( x ) k for all x ∈ G. Further more, if l ( x ) > for any x = e , one cantake ϕ to be injective, i.e., an isometric embedding. Let Z ⋊ A Z be an abelian by cyclic group, where A = (cid:20) a bc d (cid:21) ∈ GL ( Z ). We proveTheorem 0.1 by proving the following two theorems. Theorem 4.2
When the absolute value of the trace | tr( A ) | > , any length function l : Z ⋊ A Z → R ≥ vanishes on Z . Proof.
Let A = (cid:20) a bc d (cid:21) ∈ GL ( Z ) . Suppose that t is a generator of Z and t (cid:20) xy (cid:21) t − = A (cid:20) xy (cid:21) for any x, y ∈ Z . Note that (0 , t k )( v, , t k ) − = ( A k v, v ∈ Z and k ∈ Z . Therefore, an element v ∈ Z is conjugate to A k v for anyinteger k. Note that A (cid:20) (cid:21) = (cid:20) ac (cid:21) , A (cid:20) (cid:21) = (cid:20) a + bcac + dc (cid:21) and (cid:20) a + bcac + dc (cid:21) = ( a + d ) (cid:20) ac (cid:21) − ( ad − bc ) (cid:20) (cid:21) . Therefore, we have | a + d | l ( (cid:20) (cid:21) ) = l (( a + d ) (cid:20) ac (cid:21) )= l ( (cid:20) a + bcac + dc (cid:21) + ( ad − bc ) (cid:20) (cid:21) ) ≤ (1 + | ad − bc | ) l ( (cid:20) (cid:21) ) . When ad − bc = ± | a + d | > , we must have l ( (cid:20) (cid:21) ) = 0 . Similarly, we can provethat l ( (cid:20) (cid:21) ) = 0 . Since l is subadditive on Z , we get that l vanishes on Z . heorem 4.3 When the absolute value | tr( A ) | = | a + d | = 2 , I = A ∈ SL ( Z ) , anylength function l : Z ⋊ A Z → R ≥ vanishes on the direct summand of Z spanned byeigenvectors of A . Proof.
We may assume that A = (cid:20) n (cid:21) , n = 0 . For any integer k ≥ v ∈ Z , wehave t k vt − k = A k v. Take v = (cid:20) (cid:21) to get that t k (cid:20) (cid:21) t − k = (cid:20) kn (cid:21) + (cid:20) (cid:21) . Since the function l | Z is given by the norm of a Banach space according to Lemma 4.1,we get that k | n | l ( (cid:20) (cid:21) ) ≤ l ( t k (cid:20) (cid:21) t − k ) + l ( (cid:20) (cid:21) )= 2 l ( (cid:20) (cid:21) ) . Since k is arbitrary, we get that l ( (cid:20) (cid:21) ) = 0 . Remark 4.4
When A = (cid:20) (cid:21) , the semidirect product G = Z ⋊ A Z is a Heisenberggroup. A length function on G/Z ( G ) ∼ = Z gives a length function on G. In particular, alength function of G may not vanish on the second component (cid:20) (cid:21) ∈ Z < G. Remark 4.5
When A ∈ SL ( Z ) has | tr( A ) | < , the matrix A is of finite order andthe semi-direct product Z ⋊ A Z contains Z as a finite-index normal subgroup. Actually,in this case the group Z ⋊ A Z is the fundamental group of a flat -manifold M (see[61], Theorem 3.5.5). Therefore, the group Z ⋊ A Z acts freely properly discontinuouslyisometrically and cocompactly on the universal cover ˜ M = R . This means the translationlength gives a purely positive length function on Z ⋊ A Z . Lemma 4.6
Let A ∈ GL n ( Z ) be a matrix and G = Z n ⋊ A Z the semi-direct product. Let P ni =0 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 | > X j = i | a j | . Any length function l of G vanishes on Z n . Proof.
Let t be a generator of Z and tat − = Aa for any a ∈ Z n . Note that for anyinteger m, we have t m at − m = A m a and l ( a ) = l ( A m a ) . Note that n X i =0 a i A ki = 019nd thus n X i =0 a i A ki a = 0 ,l ( n X i =0 a i A ki a ) = 0for any a ∈ Z n . Therefore, | a i | l ( a ) = | a i | l ( A ki a ) = l ( X j = i a j A kj a ) ≤ X j = i | a j | l ( a ) . This implies that l ( a ) = 0 . Proof of Corollary 0.2.
When the group action is C , define l ( f ) = max { lim n → + ∞ log sup x ∈ M k D x f n k n , lim n → + ∞ log sup x ∈ M k D x f − n k n } for any diffeomorphism f : M → M. Lemma 2.5 shows that l is a length function, whichis an upper bound of the Lyapunov exponents. When the group action is Lipschitz, define L ( f ) = sup x = y d ( f x, f y ) d ( x, y )for a Lipschitz-homeomorphism f : M → M. Since L ( f g ) ≤ L ( f ) L ( g ) for two Lipschitz-homeomorphisms f, g : M → M, we have that l ( f ) := lim n →∞ max { log( L ( f n )) , log( L ( f − n )) } n gives a length function by Lemma 2.1. Note that l ( f ) ≥ h top ( f ) (see [40], Theorem 3.2.9,page 124). The vanishings of the topological entropy h top and the Lyapunov exponents inCorolary 0.2 are proved by Theorem 0.1 considering these length functions. The following lemma is a key step for our proof of the vanishing of length functions onHeisenberg groups.
Lemma 5.1
Let G = h a, b, c | aba − b − = c, ac = ca, bc = cb i be the Heisenberg group. Suppose that f : G → R is a conjugation-invariant function, i.e. f ( xgx − ) = f ( g ) for any x, g ∈ G. For any coprime integers (not-all-zero) m, n and anyinteger 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 − b − = c, we have ba − b − = a − c and thus ba − n b − = a − n c n for any integer n. Therefore, a n ba − n b − = c n and a n ba − n = c n b, a n b m a − n = a nm b m for any integer m. Thismeans [ a n , b m ] = c nm . For any coprime m, n , and any integer k, let s, t ∈ 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 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 , andany integer k. Lemma 5.2
Let G = h a, b, c | aba − b − = c, ac = ca, bc = cb i be the Heisenberg group. Any length function l : G → [0 , ∞ ) (in the sense of Definition1.1) factors through the abelization G ab := G/ [ G, G ] ∼ = Z . In other words, there is afunction l ′ : G ab → [0 , ∞ ) such that l ′ ( x n ) = | n | l ′ ( x ) for any x ∈ G ab , any integer n and l = l ′ ◦ q , where q : G → G ab is the natural quotient group homomorphism. Proof.
Let H = h c i ∼ = Z and write G = ∪ gH the union of left cosets. We choose therepresentative g ij = a i b j with ( i, j ) ∈ Z . Note that the subgroup h g ij , c i generated by g ij , c is isomorphic to Z for coprime i, j. The length function l is subadditive on h g ij , c i . By Lemma 4.1, there is a Banach space B and a group homomorphism ϕ : h g ij , c i → B such that l ( g ) = k ϕ ( g ) k for any g ∈ h g ij , c i . Lemma 5.1 implies that k ϕ ( g ij ) + ϕ ( c k ) k = k ϕ ( g ij ) k for any integer k. Since k ϕ ( g ij ) k = k ϕ ( g ij ) + ϕ ( c k ) k ≥ | k |k ϕ ( c ) k − k ϕ ( g ij ) k for any k, we have that k ϕ ( c ) k = l ( c ) = 0 . This implies that l ( g nij c m ) = k ϕ ( g nij ) + ϕ ( c m ) k = l ( g nij )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 ′ ( gH ) = l ( g ) . Since l ( g k ) = | l | l ( g ) , we have that l ′ ( g k H ) = | k | l ′ ( gH ) . The proof is finished.Denote by S ′ = { ( m, n ) | m, n are coprime integers } be the set of coprime integerpairs and define an equivalence relation by ( m, n ) ∼ ( m ′ , n ′ ) if ( m, n ) = ± ( m ′ , n ′ ). Let S = S ′ / ∼ be the equivalence classes. Theorem 5.3
Let G = h a, b, c | aba − b − = c, ac = ca, bc = cb i be the Heisenberg group.The set of all length functions l : G → [0 , ∞ ) (in the sense of Definition 1.1) is the convexpolyhedral cone R ≥ [ S ] = { X s ∈ S a s s | a s ∈ R ≥ , s ∈ S } . roof. Similar to the proof of the previous lemma, we let H = h c i ∼ = Z and write G = ∪ gH the union of left cosets. We choose the representative g ij = a i b j with ( i, j ) ∈ Z . Let T be the set of length function l : G → [0 , ∞ ) . For any length functions l, let ϕ ( l ) = P s ∈ S a s s ∈ R ≥ [ 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 ϕ : T → R ≥ [ S ] . If ϕ ( l ) = ϕ ( l ) for two functions l , l , then l ( a i b j ) = l ( a i b j ) for coprime integers i, j. Since both l , l are conjugation-invariant, Lemma 5.1 implies that l , l coincide on anycoset a i b j H and thus on the whole group G. This proves the injectivity of ϕ. For any P s ∈ S a s s, we define a function l : G = ∪ a i b j H → [0 , ∞ ) . For any coprime integers i, j, define l ( a i b j z ) = a s for any representative ( i, j ) of s andany z ∈ H. For any general integers m, n and z ∈ 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 ∈ Z . For any two elements a k b s c t , a k ′ b s ′ c t ′ we have the conjugation a k ′ b s ′ c t ′ a k b s c t ( a k ′ b s ′ c t ′ ) − = a k b s c t ′′ for some t ′′ ∈ Z . Therefore, we see that l is conjugation-invariant. The previous equalityalso shows that two elements g, h are commuting if and only if they lies simutanouslyin h a i b j , c i for a pair of coprime integers i, j. By construction, we have l ( g ) = l ( h ) . Thisproves the surjectivity of ϕ. In this section, we study length functions on matrix groups SL n ( R ). As the proofs areelementary, we present here in a separated section, without using profound results on Liegroups and algebraic groups. The following lemma is obvious. Lemma 6.1
Let G p,q = h x, t : tx p t − = x q i be a Baumslag-Solitar group. When | p | 6 = | 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 → V a unipotentlinear transformation (i.e. A k = 0 for some positive integer k ) . The following fact is fromlinear algebra (see the Lemma of page 313 in [4]. Since the reference is in Chinese, werepeat the proof here).
Lemma 6.2 I + A is conjugate to a direct sum of Jordan blocks with s along the diagonal. Proof.
We prove that V has a basis { a , Aa , · · · , A k − a , a , Aa , · · · , A k − a , · · · , a s , · · · , Aa s , · · · , A k s − a s } satisfying A k i a i = 0 for each i, which implies that the representation matrix of I + A issimilar to a direct sum of Jordan blocks with 1 along the diagonal. The proof is based22n the induction of dim V. When dim V = 1 , choose 0 = v ∈ V. Suppose that Av = λv. Then A k v = λ k v = 0 and thus λ = 0 . Suppose that the case is proved for vector spacesof dimension k < n.
Note that the invariant subspace AV is a proper subspace of V (otherwise, AV = V implies A k V = A k − V = V = 0). By induction, the subspace AV has a basis { a , Aa , · · · , A k − a , a , Aa , · · · , A k − a , · · · , a s , · · · , Aa s , · · · , A k s − a s } . Choose b i ∈ V satisfying A ( b i ) = a i . Then A maps the set { b , Ab , · · · , A k b , b , Ab , · · · , A k b , b s , · · · , Ab s , · · · , A k s b s } to the basis { a , Aa , · · · , A k − a , a , Aa , · · · , A k − a , · · · , a s , · · · , Aa s , · · · , A k s − 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 ′ s basis { b , Ab , · · · , A k − b , b , Ab , · · · , A k − b , b s , · · · , Ab s , · · · , A k s − b s , b s +1 , · · · , b s ′ } . Note that Ab i = 0 for i ≥ s + 1 and A k i +1 b i = A k i a i = 0 for each i ≤ s. This finishes theproof.
Corollary 6.3
Let A n × n be a strictly upper triangular matrix over a field K of charac-teristic ch( K ) = 2 . Then A is conjugate to A. Proof.
Suppose that A = I + u for a nilpotent matrix u. Lemma 6.2 implies that A is conjugate to a direct sum of Jordan blocks. Without loss of generality, we assume A is a Jordan block. Then A = I + 2 u + u . By Lemma 6.2 again, A is conjugate to adirect sum of Jordan blocks with 1 s along the diagonal. The minimal polynomial of A is ( x − n , which shows that there is only one block in the direct sum and thus A isconjugate to A. Recall that a matrix A ∈ GL n ( R ) is called semisimple if as a complex matrix A isconjugate to a diagonal matrix. A semisimple matrix A is elliptic (respectively, hyperbolic)if all its (complex) eigenvalues have modulus 1 (respectively, are > Lemma 6.4
Each A ∈ GL n ( R ) can be uniquely writen as A = ehu, where e, h, u ∈ GL n ( R ) are elliptic, hyperbolic and unipotent, respectively, and all three commute. The following result characterizes the continuous length functions on compact Liegroups.
Lemma 6.5
Let G be a compact connected Lie group and l a continuous length functionon G . Then l = 0 . Proof.
For any element g ∈ G, there is a maximal torus T (cid:127) g. For finite order element h ∈ 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. heorem 6.6 Let G = SL n ( R ) ( n ≥ . Let l : G → [0 , + ∞ ) be a length function,which is continuous on compact subgroups and the subgroup of diagonal matrices withpositive 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 ∈ SL n ( R ) , let g = ehu be the Jordan decomposition for commutingelements 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 ) ≤ l ( e ) + l ( h ) + l ( u ) . For any unipotent matrix u, there is an invertible matrix a such that aua − is strictlyupper triangular (see [34], Theorem 7.2, page 431). Lemma 6.3 implies that u is conjugateto 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 ) ≤ l ( h ) . Similarly, l ( h ) = l ( e − gu − ) ≤ l ( g )which implies l ( g ) = l ( h ) . Note that a hyperbolic matrix is conjugate to a real diagonalmatrix with positive diagonal entries.
Proof of Theorem 0.10.
By Theorem 6.6, the length function l is determined by itsimage on the subgroup D generated by h ( x ) , x ∈ R > . Take x = e t , t ∈ R . We have l ( h ( e kl )) = | k || l | l ( h ( e )) for any rational number kl . Since l satisfies the condition 2) of thedefinition and is continuous on D , we see that l | D is determined by the image l ( h ( e ))(actually, any real number t is a limit of a rational sequence). Note that the translationfunction τ vainishes on compact subgroups and is continuous on the subgroup of diagonalsubgroups with positive diagonal entries (cf. [16], Cor. 10.42 and Ex. 10.43, page 320).Therefore, l is proposional to τ . Actually, τ can be determined explicitly by the formulatr( A ) = ± τ ( A )2 (for nonzero τ ( A )), where tr is the trace and cosh is the hyperboliccosine function (see [5], Section 7.34, page 173). This implies that l ( A ) is determined bythe spectrum radius of A (which could also be seen clearly by the matrix norm).Let h i ( x ) ( i = 2 , · · · , n ) be an n × n diagonal matrix whose (1 , x, ( i, i )-entry is x − , while other diagonal entries are 1s and non-diagonal entries are 0s. Thesubgroup D < SL n ( R ) of diagonal matrices with positive diagonal entries is isomorphicto ( R > ) n − and D is generated by the matrices h i ( x ) ( i = 2 , · · · , n ) whose (1 , x, ( i, i )-entry is x − . Since h i ( x ) ( i = 1) is conjugate to h ( x ) , a length function l : SL n ( R ) → [0 , + ∞ ) is completely determined by its image on the convex hull spannedby h ( e ) , h ( e ) , · · · , h n ( 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 : SL ( R ) → [0 , + ∞ ) be a non-trivial length function that is contin-uous on the subgroup SO (2) and the diagonal subgroup. Then l ( g ) > if and only if g ishyperbolic. Proof.
It is well-known that the elements in SL ( R ) are classified as elliptic, hyperbolicand parabolic elements. Moreover, the translation length τ vanishes on the compactsubgroup SO (2) and the parabolic elements. The corollary follows Theorem 0.10.When the length function l is the asymptotic distortion function dist ∞ , Corollary 6.7is known to Navas [24] (Proposition 4). 24 Length functions on algebraic and Lie groups
For an algebraic group G, let k [ G ] be the regular ring. For any g ∈ G, let ρ g : k [ G ] → 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 ∈ G. There exists uniqueelements g s , g u such that g s g u = g u g s , and ρ g s is semisimple, ρ g u is unipotent. Lemma 7.2 [46] Let G be a reductive connected algebraic group over an algebraicallyclosed 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 afield k and g ∈ G ( k ) then the set of conjugacy classes in G ( k ) which when base changedto the algebraic closed field ¯ k are equal to the conjugacy class of g in G (¯ k ) is in bijectionwith the subset of H (¯ 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 manyextensions of k of degree n (in a fixed algebraic closure ¯ k of k ). Examples of fields of type (F) include: the field R of reals, a finite field, the field offormal 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 alinear algebraic group defined over k . The set H (¯ 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 ) ≤ l ( g s ) + l ( g u ) and l ( g s ) ≤ l ( g ) + l ( g − u ) . Note that for any integer n, g nu is also unipotent. By the Lemma 7.2,Lemma 7.3 and Lemma 7.5, there are only finitely many conjugacy classes of unipotentelements. This implies that g n u = g n u for distinct positive integers n , n . Therefore, wehave n l ( g u ) = n 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 istrivial. Let g be its Lie algebra and let exp : g → G denote the exponential map. Anelement x ∈ g is real semi-simple if Ad ( x ) is diagonalizable over R . An element g ∈ G iscalled 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 wewrite x = log g . The following is the Jordan decomposition in Lie groups. An element e ∈ G is elliptic if Ad ( e ) is diagonalizable over C with eigenvalues 1. Lemma 7.7 ([42], Prop. 2.1 and Remark 2.1)1. Let g ∈ 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 threeelements e ( g ) , h ( g ) , u ( g ) commute. . An element f ∈ G commutes with g if and only if f commutes with the threecomponents. Moreover, if f, g commutes, then e ( f g ) = e ( f ) e ( g ) , h ( f g ) = h ( f ) h ( g ) , u ( f g ) = u ( f ) u ( g ) . Lemma 7.8 [Eberlein, Prop. 1.14.6, page 63]Let G be a connected semisimple Lie groupwhose center is trivial. Then there exists an integer n ≥ and an algebraic group G ∗ < GL n ( C ) defined over Q such that G is isomorphic to G ∗ R (the connected component of G ∗ R containing the identity) as a Lie group. Let G = KAN be an Iwasawa decomposition. The Weyl group W is the finite groupdefined as the quotient of the normalizer of A in K modulo the centralizer of A in K. Foran element x ∈ 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 ∈ G is hyperbolic if and only if it is conjugateto 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 anIwasawa decomposition G = KAN . Let W be the Weyl group.(i) Any length function l on G that is continuous on the maximal compact subgroup K is determined by its image on A. (ii) Conversely, any length function l on A that is W -invariant (i.e. l ( w · a ) = l ( a ) )can be extended to be a length function on G that vanishes on the maximal compactsubgroup K. Proof. (i) Let Z be the center of G. Then
G/Z is connected with trivial center. For any z ∈ Z, g ∈ G, we have l ( z ) = 0 and l ( gz ) = l ( g ) . The length function l factors through alength function on G/Z.
We may assume that G has the trivial center. For any g ∈ G, theJordan decomposition gives g = ehu, where e is elliptic, h is hyperbolic and u is unipotentand where the three elements e, h, u commute (cf. Lemma 7.7). By Lemma 7.8, the Liegroup G is an algebraic group. Lemma 7.6 implies that l vanies on unipotent elementsand 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 ∈ G, a ∈ A, define l ′ ( gag − ) = l ( a ) . If g a g − = g a g − for g , g ∈ G, a , a ∈ A, then g − g a g − g = a . By Lemma 7.9, there exists an element w ∈ W such that w · a = a . Therefore, we have l ( a ) = l ( a ) and thus l ′ is well-defined onthe set H of conjugates of elements in A. Such a set H is the set consisting of hyperbolicelements by Lemma 7.9. We then extend l ′ on the set of all conjugates of elements in K. For any g ∈ G, k ∈ K, define l ′ ( gkg − ) = 0 . If g kg − = g ag − for g , g ∈ G, k ∈ K, a ∈ A, then g − g ag − g = 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 ′ ( g kg − ) = l ( g ag − ) = 1 . l ′ is well-defined on the set of hyperbolic elements and elliptic elements.For any unipotent element u ∈ G, define l ′ ( u ) = 0 . For any element g ∈ G, let g = ehu be the Jordan decomposition. Define l ′ ( g ) = l ′ ( h ) . We check the function l ′ is a length function on G. The definition shows that l ′ isconjugate invariant. For any positive integer n and any g ∈ G with Jordan decomposition g = ehu, we have g n = e n h n u n and thus l ′ ( g n ) = l ′ ( h n ) . But h n is hyperbolic andconjugate to an element in A (see Lemma 7.9). Therefore, we have l ′ ( h n ) = | n | l ′ ( h ) andthus l ′ ( g n ) = | n | l ′ ( g ) . If g = e h u commutes with g = e h u , then g g = e e h h u u (cf. Lemma 7.7) and l ′ ( g g ) = l ′ ( h h ) . Since h , h are commuting hyperbolic elements,they are conjuate simutaniously to elements in A. Therefore, we have l ′ ( h h ) ≤ l ′ ( h ) + l ′ ( h )and l ′ ( g g ) ≤ l ′ ( g ) + l ′ ( g ) . Remark 7.11
A length function l on A is determined by a group homomorphism f : A → B , for a real Banach space ( B , kk ) , satisfying l ( a ) = k f ( a ) k (see Lemma 4.1). Theprevious theorem implies that a length function l on the Lie group G (that is continuouson compact subgroup) is uniqely determined by such a group homomorphism f : A → B such that k f ( a ) k = k f ( wa ) k for any a ∈ A and w ∈ W, the Weyl group. Let G be a connected semisimple Lie group whose center is finite with an Iwasawadecomposition G = KAN . Let exp : g → G be the exponent map from the Lie algebra g with subalgebra 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 ∈ h ) in a fixed closed Weylchamber of A. Proof.
Let Z be the center of G. Then
G/Z is connected with trivial center. The lengthfunction l factors through an length function on G/Z (cf. Corrolary 1.8). For any g ∈ G we have g = ehu, where e is elliptic, h is hyperbolic and u is unipotent and where thethree 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 ∈ A is conjugate to an element in a fixed Weyl chamber C (cf. [38], Theorem 8.20, page254). For any element exp( x ) ∈ C, with unit vector x ∈ h , the one-parameter subgroupexp( R x ) lies in A. Since l is continuous on A, the function l is determined by its image onexp( Q x ) . Note that l ( e mn x ) = n l ( e mx ) = mn l ( e x ) for any rational number mn . 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 realrank . There is essentially only one length function on G. In order words, any continuouslength 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 itsimage on a unit vector in a split torus. 27 Rigidity of group homomorphisms on arithmeticgroups
Let V denote a finite-dimensional vector space over C , endowed with a Q -structure. Recallthat the arithmetic subgroup is defined as the following (cf. Borel [11], page 37). Definition 8.1
Let G be a Q -subgroup of GL( V ) . A subgroup Γ of G Q is said to bearithmetic if there exists a lattice L of V Q such that Γ is commensurable with G L = { g ∈ G Q : gL = L } . Theorem 8.2
Let Γ be an arithmetic subgroup of a simple algebraic Q -group of Q -rankat least . Suppose that H is a group with a purely positive length function. Then anygroup homomorphism f : Γ → H has its image finite. Recall that a group G is quasi-simple, if any non-trivial normal subgroup is eitherfinite or of finite index. The Margulis-Kazhdan theorem (see [62], Theorem 8.1.2) impliesthat an irreducible lattice (and hence) in a semisimple Lie group of real rank ≥ Lemma 8.3
Let Γ be a finitely generated quasi-simple group with contains a Heisen-berg subgroup, i.e. there are elements torsion-free elements a, b, c ∈ Γ satisfying [ a, b ] = c, [ a, c ] = [ c, b ] = 1 . Suppose that G has a virtually poly-positive length. Then any grouphomomorphism f : Γ → G has its image finite. Proof.
Suppose that G has a finite-index subgroup H and a subnormal series1 = H n ⊳ H n − ⊳ · · · ⊳ H = H such that every finitely generated subgroup of H i /H i +1 has a purely positive length func-tion. Without loss of generality, we assume that H is normal. Let f : Γ → G be ahomomorphism. The kernel of the composite f : Γ f → G → G/H is finitely generated. Suppose that the image of the composite f : ker f f → H → H/H has a purely positive length function l. After passing to finite-index subgroups, we maystill suppose that ker f contains a Heisenberg subgroup h a, b, c i . By Lemma 5.2, the lengthfunction l vanishes on f ( c ) . Therefore f ( c k ) = 1 ∈ H/H for some integer k > . Thenormal subgroup ker f containing c k is of finite index. Now we have map ker f → H induced by f. An induction argument shows that f maps some power c d of the centralelement of the Heisenberg subgroup into the identity 1 ∈ G. Therefore, the image of f isfinite. Proof of Theorem 8.2.
It is well-known that that G contains a Q -split simple subgroupwhose root system is the reduced subsystem of the Q -root system of G (see [13], Theorem7.2, page 117). Replacing G with this Q -subgroup, we may assume G is Q -split and theroot system of G is reduced. Because G is simple and Q -rank( G ) ≥
2, we know thatthe Q -root system of G is irreducible and has rank at least two. Therefore, the Q -root28ystem of G contains an irreducible subsystem of rank two, that is, a root subsystem oftype A , B , G (see [60], page 338). For A , choose { α , α } as a set of simple roots (seeFigure 1).Then the root element x α + α ( rs ) = x α ( rs ) is a commutator [ x α ( r ) , x α ( s )] , with x α ( rs )commutes with x α ( r ) , x α ( s ) . For G , the long roots form a subsystem of A . For B , choose { α , α } as a set of simple roots (see Figure 2). The long root element x α (2 rs )is a commutator [ x α ( r ) , x α ( s )] of the two short root elements, and x α (2 rs ) commuteswith x α ( r ) , x α ( s ) (cf. [37], Proposition of page 211). This shows that the arithmeticsubgroup Γ 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 con-sider the stable word lengths, the following is essentially already known (cf. Polterovich[53], Corollary 1.1.D and its proof). Proposition 8.4
Let Γ be an irreducible non-uniform lattice in a semisimple connected,Lie group without compact factors and with finite center of real rank ≥ . Assume that agroup G has a virtually poly-positive stable word length. In other words, the group G hasa finite-index subgroup H and a subnormal series H n ⊳ H n − ⊳ · · · ⊳ H = H such that every finitely generated subgroup of H i /H i +1 ( i = 0 , , · · · , n − ) has a purelypositive stable word length. Then any group homomorphism f : Γ → G has its imagefinite. Proof.
Without loss of generality, we assume that f takes image in H. Since a lattice isfinitely generated, Γ has its image in H /H finitely generated. When the image has apurely positive word length, any distorted element in Γ must have trivial image in H /H (see ). Lubotzky, Mozes and Raghunathan [44] prove that irreducible non-uniform latticesin higher rank Lie groups have non-trivial distortion elements (They actually prove thestronger result that there are elements in the group whose word length has logarithmic29rowth). Then a finite-index subgroup Γ < Γ will have image in H , since high-rankirreducible lattices are quasi-simple. An induction argument finishes the proof.When we consider the length given by quasi-cocyles, the following is also essentiallyalready known ( cf. Py [54], Prop. 2.2, following Burger-Monod [18] [19]). Recall thata locally compact group has property (TT) if any continuous rough action on a Hilbertspace has bounded orbits (see [49], page 172). Burger-Monod proves that an irreduciblelattice Γ in a high-rank semisimple Lie group has property (TT). Proposition 8.5
Let Γ be an irreducible lattice in a semisimple connected, Lie groupwithout compact factors and with finite center of real rank ≥ . . 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 H n ⊳ H n − ⊳ · · · ⊳ H = H such that every finitely generated subgroup of H i /H i +1 ( i = 0 , , · · · , n − ) has a purelypositive length given by a quasi-cocycle with values in Hilbert spaces. Then any grouphomomorphism f : Γ → G has its image finite. Proof.
Note that a group Γ has property (TT) if and only if H (Γ; E ) = 0 andker( H b (Γ; E ) → H (Γ; E )) = 0for any linear isometric action of Γ on a Hilbert space E. Here H b (Γ; E ) is the secondbounded cohomology group. Suppose that u : Γ → E is a quasi-cocyle. There is abounded map v : Γ → E and a 1-cocycle w : Γ → E such that u = v + w, by Proposition 2.1 of Py [54]. Since Γ has property T, there exists x ∈ E such that w ( γ ) = γx − x . Therefore, we have k u ( γ n ) k n = k v ( γ n ) + w ( γ n ) k n = k v ( γ n ) + γ n x − x k n ≤ k v ( γ n ) k + 2 k x k n → . Without loss of generality , we assume that G = H. Suppose that any finitely generatedsubgroup of
H/H has a purely positive average norm l given by a cocycle. The compositeΓ f → H → H/H has a finite-index kernel Γ , since l vanishes on infinite-order elements of the image. Thisimplies that f (Γ ) lies in H . A similar argument proves that ker f is of finite index inthe general case. 30 Rigidity of group homomorphisms on matrix groups
Recall that a ring R is right Artinian if any non-empty family of right ideals containsminimal elements. A ring R is semi-local if R/ rad( R ) is right Artinian (see Bass’ K-theory book [3] page 79 and page 86), where rad( R ) is the Jacobson radical. Let n bea positive integer and R n the free R -module of rank n with standard basis. A vector( a , . . . , a n ) in R n is called right unimodular if there are elements b , . . . , b n ∈ R suchthat a b + · · · + a n b n = 1. The stable range condition sr m says that if ( a , . . . , a m +1 )is a right unimodular vector then there exist elements b , . . . , b m ∈ R such that ( a + a m +1 b , . . . , a m + a m +1 b m ) is right unimodular. It follows easily that sr m ⇒ sr n for any n ≥ m . A semi-local ring has the stable range sr ( [3], page 267, the proof of Theorem9.1). A finite ring R is right Artinian and thus has sr . The stable rangesr( R ) = min { m : R has sr m +1 } . Thus sr( R ) = 1 for a finite ring R .We briefly recall the definitions of the elementary subgroups E n ( R ) of the generallinear group GL n ( R ), and the Steinberg groups St n ( R ). Let R be an associative ring withidentity and n ≥ n ( R ) is the group of all n × n invertible matrices with entries in R . For an element r ∈ R and any integers i, j such that 1 ≤ i = j ≤ n, denote by e ij ( r ) the elementary n × n matrix with 1 s in thediagonal 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 ) = h e ij ( r ) | ≤ i = j ≤ n, r ∈ R i . Denote by I n the identity matrix and by [ a, b ] the commutator aba − b − . 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 ∈ R. Then for distinct integers i, j, k, l with ≤ i, j, k, l ≤ n, the following hold:(1) e ij ( r + s ) = e ij ( r ) e ij ( s ); (2) [ e ij ( r ) , e jk ( s )] = e ik ( rs ); (3) [ e ij ( r ) , e kl ( s )] = I n . By Lemma 9.1, the group E n ( R ) ( n ≥
3) is finitely generated when the ring R isfinitely generated. Moreover, when n ≥ , the group E n ( R ) is normally generated by anyelementary matrix e ij (1) . The commutator formulas can be used to define Steinberg groups as follows. For n ≥ , the Steinberg group St n ( R ) is the group generated by the symbols { x ij ( r ) : 1 ≤ i = j ≤ n, r ∈ R } subject to the following relations:(St1) x ij ( r + s ) = x ij ( r ) x ij ( s );(St2) [ x ij ( r ) , x jk ( s )] = x ik ( rs ) for i = k ; 31St3) [ x ij ( r ) , x kl ( s )] = 1 for i = l, j = k. There is an obvious surjection St n ( R ) → E n ( R ) defined by x ij ( r ) e ij ( r ) . For any ideal I ⊳ R, let p : R → R/I be the quotient map. Then the map p inducesa group homomorphism p ∗ : St n ( R ) → St n ( R/I ) . Denote by St n ( R, I ) ( resp., E n ( R, I ))the subgroup of St n ( R ) ( resp., E n ( R )) normally generated by elements of the form x ij ( r )( resp., e ij ( r )) for r ∈ I and 1 ≤ i = j ≤ n. In fact, St n ( R, I ) is the kernel of p ∗ (cf.Lemma 13.18 in Magurn [47] and its proof). However, E n ( R, I ) may not be the kernel of E n ( R ) → E n ( R/I ) induced by p. Lemma 9.2
When n ≥ sr( R ) + 2 , the natural map St n ( R ) → St n +1 ( R ) is injective. Inparticular, when R is finite, the Steinberg group St n ( R ) is finite for any n ≥ . Proof.
Let W ( n, R ) be the kernel of the natural map St n ( R ) → St n +1 ( R ) . When n ≥ 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 St n ( R ) is the universal central extension of E n ( R ) (cf. [59], Proposition 5.5.1. page 240). Therefore, the kernel St n ( R ) → E n ( R ) is thesecond homology group H ( E n ( R ); Z ) . When R is finite, both E n ( R ) and H ( E n ( R ); Z )are finite. Therefore, the group St n ( R ) is finite for any n ≥ . Theorem 9.3
Suppose that G is a group satisfying that1) G has a purely positive length function, i.e. there is a length function l : G → [0 , ∞ ) such that l ( g ) > for any infinite-order element g ; and2) any torsion abelian subgroup of G is finitely generated.Let R be an associative ring with identity and St n ( R ) the Steinberg group. Suppose that S < St n ( R ) is a finite-index subgroup. Then any group homomorphism f : St n ( R ) → G has its image finite when n ≥ . Proof.
Since any ring R is a quotient of a free (non-commutative) ring Z h X i for someset X and St n ( R ) is functorial with respect to the ring R, we assume without loss ofgenerality that R = Z h X i . We prove the case S = St n ( R ) first. Let x ij = h x ij ( r ) : r ∈ R i , which is isomorphic to the abelian group R. Note that[ x (1) , x (1)] = x (1)and x (1) commutes with x (1) and x (1) . Lemma 5.2 implies any length function van-ishes on x (1) . By Lemma 1.4, the length l ( f ( x (1))) = 0 . Note that x ij ( r ) is conjugateto x ( r ) for any r ∈ R and i, j satisfying 1 ≤ i = j ≤ n. Since l is purely postive, we getthat f ( x (1)) is of finite order. Let I = ker f | x . Then I = ∅ , as f ( x (1)) is of finiteorder. For any x ∈ I, and y ∈ R, we have x ( xy ) = [ x ( x ) , x ( y )] . Therefore, f ( x ( xy )) = [ f ( x ( x ) , f ( x ( y )))] = 1and thus xy ∈ I. Similarly, we have f ( x ( yx )) = f ([ x ( y ) , x ( x )]) = 1 . This provesthat I is a (two-sided) ideal. Note that f ( e ) = R/I is a torsion abelian group. By the32ssumption 2), the quotient ring
R/I is finite. Let St n ( R, I ) be the normal subgroup ofSt n ( R ) generated by x ij ( r ) , r ∈ I. There is a short exact sequence1 → St n ( R, I ) → St n ( R ) → St n ( R/I ) → . Since
R/I is finite, we know that St n ( R/I ) is finite by Lemma 9.2. This proves that Im f is finite since f factors through St n ( R/I ). For general finite-index subgroup S, we assume S is normal in St n ( R ) after passing to a finite-index subgroup of S. A similar proof showsthat S contains St n ( R, I ) for some ideal I with the quotient ring R/I finite. Therefore,the image Im 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 finiteindex (eg. the ring of algebraic integers in a number field). Suppose that S < St n ( R ) is afinite-index subgroup of the Steinberg group. Then any group homomorphism f : S → G has its image finite when n ≥ . Proof.
The proof is similar to that of Theorem 9.3. Let I = ker f | x , where x = S ∩ h x ( r ) : r ∈ R i . Since R is of characteristic zero and the length l ( f ( x ( k ))) = 0 forsome integer k, we have f ( x ( k )) is of finite order. Therefore, f ( x ( k ′ )) = 1 for someinteger k ′ , which proves that the ideal I is nozero. Since I is of finite index in R, we getthat St n ( R, I ) is of finite index in S. This finishes the proof.Since the natural map St n ( R ) → E n ( R ) is surjective, any group homomorphism f : E n ( R ) → G can be lifted to be a group homomorphism St n ( R ) → G. Moreover, a finite-index subgroup E of E n ( R ) is lifted to be a finite-index subgroup S of St n ( R ) . Theorem0.4 and Theorem 0.6 follows Theorem 9.3 and Theorem 9.4, by inductive arugments onthe 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 checkthe two conditions for G in Theorem 0.4. Lemma 3.6 proves that G has a purely positivelength function. When G is a CAT(0) group, (i.e. G acts properly and cocompactlyon a CAT(0) space), then any solvable subgroup of G is finitely generated (and actuallyvirtually 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 offinite subgroups and thus a torsion abelian subgroup is finite (see [16], Theorem 3.2, page459). Birman-Lubotzky-McCarthy [7] proves that any abelian subgroup of the mappingclass groups for orientable surfaces is finitely generated. Bestvina-Handel [6] proves thatevery solvable subgroup of Out( F k ) has a finite index subgroup that is finitely generatedand free abelian. When G is the diffeomorphism group Diff(Σ , ω ) , there is a subnormalseries (see the proof of Lemma 3.8)1 ⊳ Ham(Σ , ω ) ⊳ Diff (Σ , ω ) ⊳ Diff(Σ , ω ) , with subquoitents in Ham(Σ , ω ) , H (Σ , R ) and the mapping class group MCG(Σ) . Anyabelian 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 forevery group G acting properly on a CAT(0) (or a Gromov hyperbolic) space. We don’tknow whether the condition 2) can be dropped. Let k be a field and k ( x , x , · · · , x n ) be the field of rational functions in n indeterminatesover k. It is well-known that the Cremona group Cr n ( k ) is isomorphic to the automorphismgroup Aut k ( k ( x , x , · · · , x n )) of the field k ( x , x , · · · , x n ). Lemma 10.1
Let f : k ( x , x , · · · , x n ) → k ( x , x , · · · , x n ) be given by f ( x ) = αx , f ( x i ) = x i for some = α ∈ k and any i = 2 , · · · , n. Then f lies in the center of a Heisenberg sub-group. In other words, there exists g, h ∈ Cr n ( k ) such that [ g, h ] = ghg − h − = f, [ g, f ] =1 and [ h, f ] = 1 . Proof.
Let g, h : k ( x , x , · · · , x n ) → k ( x , x , · · · , x n ) be given by g ( x ) = x x , g ( x i ) = x i ( i = 2 , · · · , n )and h ( x ) = x , h ( x ) = α − x , h ( x j ) = x j ( j = 3 , · · · , n ) . It can be directly checked that [ g, h ] = f, [ g, f ] = 1 and [ h, f ] = 1 . Lemma 10.2
Let l : Bir( P nk ) → [0 , ∞ ) be a length function ( n ≥ . Then l vanishes ondiagonal elements and unipotent elements of Aut( P nk ) = PGL n +1 ( k ) . Proof.
Let g = diag( a , a , · · · , a n ) ∈ 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 enoughto prove that l (diag(1 , · · · , , a i , , · · · , , where diag(1 , · · · , , a i , , · · · ,
1) is thediagonal matrix with a i in the ( i, i )-th position and all other diagonal entries are 1 . But diag(1 , · · · , , a i , , · · · ,
1) is conjugate to diag(1 , α, , · · · ,
1) for α = a i . Lemma10.1 implies that diag(1 , α, , · · · ,
1) lies in the center of a Heisenberg group. Therefore, l (diag(1 , α, , · · · , l ( g ) = 0 . The vanishing of l onunipotent elements follows Corollary 6.3 when the characteristic of k is not 2. When thecharacteristic of k is 2 , any unipotent element A = I + u (where u is nilpotent) is of finiteorder. This means l ( A ) = 0. Proof of Theorem 0.8.
When k is algebraically closed, the Jordan normal form impliesthat any element g ∈ PGL n ( k ) is conjugate to the form sn with s diagonal and n thestrictly upper triangular matrix. Moreover, sn = ns. Therefore, l ( f ) ≤ l ( s ) + l ( n ) . ByLemma 10.2, l ( s ) = l ( n ) = 0 and thus l ( g ) = 0. Proof of Corollary 0.9.
Let f : Bir( P k ) → G be a group homomorphism. Supposethat G has a purely positive length function l. By Theorem 0.8, the purely positive lengthfunction l on G will vanish on f (PGL ( k )) . Since k is infinite and PGL ( k ) is a simplegroup, we get that PGL ( k ) lies in the ker f. By Noether-Castelnuovo Theorem, Bir( P k )is generated by PGL ( k ) and an involution. Moreover, the Bir( P k ) is normally generatedby PGL ( k ) . Therefore, the group homomorphism f is trivial. The general case is provedby an inductive argument on the subnormal series of a finite-index subgroup of G. Lemma 10.3
Let
Bir( P n R ) ( n ≥ be the real Cremona group. Any length function l :Bir( P nk ) → [0 , ∞ ) , which is continuous on PSO( n +1) < Aut( P n R ) , vanishes on PGL n +1 ( R ) . roof. By Lemma 10.2, the length function l vanishes on diagonal matrices of PGL n +1 ( R ) . Theorem 0.11 implies that l vanishes on the whole group PGL n +1 ( R ) . Acknowledgements
The author wants to thank many people for helpful discussions, including WenyuanYang on a discussion of hyperbolic groups, C. Weibel on a discussion on Steinberg groupsof finite rings, Feng Su on a discussion of Lie groups, Ying Zhang on a discussion oftranslation lengths of hyperbolic spaces, Enhui Shi for a discussion on smooth measure-theoretic entropy.
References [1] E. Alibegovi´c,
Translation lengths in
Out( F n ), Geom. Dedicata 92 (2002), 87–93.[2] W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature ,Progress in Mathematics, vol. 61, Birkh¨auser Boston Inc., Boston, MA, 1985.[3] H. Bass,
Algebraic K-Theory , Benjamin, New York, 1968.[4] Algebra Group of Beida,
Gao Deng Dai Shu (Advanced algebra) , High educationpress, 2013.[5] A. Beardon,
The geometry of discrete groups , Graduate Texts in Math., Vol. 91,Springer-Verlag, New York, 1983.[6] M. Bestvina, M. Handel,
Solvable Subgroups of
Out( F n ) are Virtually Abelian , Ge-ometriae Dedicata volume 104, pages 71–96 (2004).[7] J. S. Birman, A. Lubotzky, and J. McCarthy, Abelian and solvable subgroups of themapping class groups , Duke Math. J. 50 (1983), no. 4, 1107–1120.[8] J. Blanc and S. Cantat,
Dynamical degrees of birational transformations of projectivesurfaces , J. Amer. Math. Soc. 29 (2016), 415-471.[9] J. Blanc and J.-P. Furter,
Length in the Cremona group , Ann. H. Lebesgue 2 (2019),p.187-257.[10] J. Blanc, S. Lamy and S. Zimmermann,
Quotients of higher dimensional Cremonagroups , Acta Math. (to appear), arXiv:1901.04145.[11] A. Borel,
Introduction to Arithmetic groups , (translated by L. Pham and translationedited by D. Morris) American Math. Soc., 2019.[12] A. Borel, J.-P. Serre,
Th´eor`emes de finitude en cohomologie galoisienne , Comment.Math. Helv. 39 (1964) 111–164 (in French).[13] A. Borel and J. Tits,
Groupes r´eductifs,
Inst. Hautes ´Etudes. Sci. Publ. Math. 27(1965), 55-150.[14] M.R. Bridson,
The rhombic dodecahedron and semisimple actions of
Aut ( F n ) on CAT (0) spaces , Fund. Math. 214 (2011).[15] M.R. Bridson,
Length functions, curvature and the dimension of discrete groups ,Mathematical Research Letters 8, 557–567 (2001).[16] M.R. Bridson and A. Haefliger,
Metric spaces of nonpositive curvature , Grundlehrender Math. Wiss. 319, Springer-Verlag, Berlin, 1999.3517] M. Bridson, R. Wade,
Actions of higher-rank lattices on free groups.
Compos. Math.147, 1573–1580 (2011).[18] M. Burger and N. Monod,
Bounded cohomology of lattices in higher rank Lie groups ,J. Eur. Math. Soc. (JEMS) 1, No. 2 (1999), 199–235.[19] M. Burger and N. Monod,
Continuous bounded cohomology and applications to rigid-ity theory , Geom. Funct. Anal. 12, No. 2 (2002), 219–280.[20] D. Calegari, scl , MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009.[21] S. Cantat,
The Cremona group in two variables.
Proceedings of the sixth EuropeanCongress of Math., 187:211–225, 2013.[22] D. Calegari, M. Freedman,
Distortion in transformation groups , Geometry & Topol-ogy 10 (2006) 267–293.[23] M. Ershov, A. Jaikin-Zapirain,
Property (T) for noncommutative universal lattices ,Invent. Math. 179 (2010) 303–347.[24] H. Eynard-Bontemps and A. Navas,
Mather invariant, distortion, and conjugates fordiffeomorphisms of the interval , arXiv:1912.09305.[25] B. Farb, A. Lubotzky and Y. N Minsky,
Rank-1 phenomena for mapping class groups ,Duke Mathematical Journal, Volume 106, Number 3, 581–597, 2001.[26] B. Farb, H. Masur,
Superrigidity and mapping class groups , Topology 37, 1169–1176,1998.[27] J. Franks and M. Handel,
Area preserving group actions on surfaces , Geom. Topol.7, (2003), 757–771.[28] T. Fritz, S. Gadgil, A. Khare, P. Nielsen, L. Silberman and T. Tao,
Homogeneouslength functions on groups , Algebra and Number Theory, Vol. 12 (2018), No. 7,1773–1786.[29] S. Garge and A. Singh,
Finiteness of z-classes in reductive groups , Journal of Algebra554 (2020) 41–53.[30] S.M. Gersten,
A presentation for the special automorphism group of a free group , J.Pure Appl. Algebra, 33 (1984) 269-279.[31] S.M. Gersten,
The automorphism group of a free group is not a CAT(0) group , Proc.Amer. Math. Soc. 121 (1994), 999-1002.[32] M. Gromov,
Hyperbolic groups , in “Essays in Groups Theory” (S. Gersten, ed.),MSRI Publications 5 (1989) 75–263.[33] T. Haettel, Hyperbolic rigidity of higher rank lattices (with appendix by VincentGuirardel and Camille Horbez), arXiv:1607.02004.[34] S. Helgason,
Differential geometry, Lie groups and symmetric spaces,
AcademicPress, New York, 1978.[35] H. Hu, Some ergodic properties of commuting diffeomorphisms. Ergodic Theory Dy-nam. Systems 13 (1993), no. 1, 73–100.[36] H. Hu, E. Shi, Z. Wang,
Some ergodic and rigidity properties of discrete Heisenberggroup actions , Israel Journal of Mathematics 228 (2018), pages 933–972.[37] J. Humphreys,
Linear algebraic groups , Springer, 1975.3638] J. Humphreys,
Introduction to Lie algebras and representation theory , Springer-Verlag, New York, 1972.[39] A. Kaimanovich, H. Masur,
The Poisson boundary of the mapping class group . Invent.Math. 125, 221–264 (1996).[40] A. Katok and B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Sys-tems , Encyclopedia of Mathematics and Its Applications 54, Cambridge Univer-sity Press, 1995.[41] M. Kolster,
On injective stability for K , Proceedings, Oberwolfach, 1980,in “Lecture Notes in Mathematics No. 966,” pp. 128-168, Springer-Verlag,Berlin/Heidelberg/New York, 1982.[42] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition , AnnalesScientifiques de l’ ´Eole Normale Sup´erieure, S´erie 4, 6(1973): 413–455.[43] G. Levitt,
Counting growth types of automorphisms of free groups . Geometric andFunctional Analysis, 19(4):1119–1146, 2009.[44] A. Lubotzky, S. Mozes and M.S. Raghunathan,
The word and riemannian metricson lattices in semisimple Lie groups , IHES Publ. Math. 91 (2000), 5-53.[45] A. Lonjou and C. Urech,
Actions of Cremona groups on CAT(0) cube complexes ,arXiv:2001.00783v1.[46] G. Lusztig,
On the finiteness of the number of unipotent classes , Invent. Math. 34(3) (1976) 201–213.[47] B.A. Magurn.
An algebraic introduction to K-theory,
Cambridge University Press,2002.[48] M. Mimura,
Superrigidity from Chevalley groups into acylindrically hyperbolic groupsvia quasi-cocycles , Journal of the European Mathematical Society, Volume 20,Issue 1, 2018, pp. 103-117.[49] N. Monod, Continuous bounded cohomology of locally compact groups, LectureNotes in Mathematics 1758, Springer-Verlag, Berlin, (2001).[50] A. Navas,
On conjugates and the asymptotic distortion of 1-dimensional C bv dif-feomorphisms , arXiv:1811.06077.[51] A. Navas, Groups of Circle Diffeomorphisms , Univ. Chicago Press, Chicago, 2011.[52] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbersfor dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197-221.[53] L. Polterovich,
Growth of maps, distortion in groups and symplectic geometry , Invent.Math. 150, No. 3 (2002), 655–686.[54] P. Py,
Some remarks on area-preserving actions of lattices , in Geometry, Rigidity andGroup Actions, The University of Chicago Press, Chicago and London (2011).[55] J-P Serre,
Trees, Springer Monographs in Mathematics , Springer, Berlin (2003).Translated from the French original by John Stillwell, Corrected 2nd printingof the 1980 English translation.[56] M. Steele,
Probability theory and combinatorial optimization , SIAM, Philadelphia(1997). 3757] A. Thom,
Low degree bounded cohomology and L2-invariants for negatively curvedgroups , Groups Geom. Dyn. 3 (2009), 343–358.[58] P. Walters,
An Introduction to Ergodic Theory , Springer-Verlag, New York, 1981.[59] C. Weibel,
The K-book: an introduction to algebraic K-theory.
Graduate studies inmathematics, 145, American mathematical society, Providence, 2013.[60] D. Witte,