aa r X i v : . [ m a t h . G R ] F e b On uniformly continuous endomorphisms of hyperbolicgroups
Andr´e Carvalho ∗ Centre of Mathematics, University of Porto, R. Campo Alegre, 4169-007 Porto, Portugal
Abstract
We prove a generalization of the fellow traveller property for a certain typeof quasi-geodesics and use it to present three equivalent geometric formulationsof the bounded reduction property. We then provide an affirmative answerto a question from Ara´ujo and Silva as to whether every nontrivial uniformlycontinuous endomorphism of a hyperbolic group with respect to a visual metricsatisfies a H¨older condition. We remark that these results combined with thework done by Paulin prove that every endomorphism admitting a continuousextension to the completion has a finitely generated fixed point subgroup.
The dynamical study of endomorphisms of groups started with the (independent)work of Gersten [12] and Cooper [8], using respectively graph-theoretic and topo-logical approaches. They proved that the subgroup of fixed points Fix( ϕ ) of somefixed automorphism ϕ of F n is always finitely generated, and Cooper succeeded onclassifying from the dynamical viewpoint the fixed points of the continuous exten-sion of ϕ to the boundary of F n . Bestvina and Handel subsequently developed thetheory of train tracks to prove that Fix( ϕ ) has rank at most n in [2]. The problemof computing a basis for Fix( ϕ ) had a tribulated history and was finally settled byBogopolski and Maslakova in 2016 in [3].This line of research extended early to wider classes of groups. For instance,Paulin proved in 1989 that the subgroup of fixed points of an automorphism ofa hyperbolic group is finitely generated [16]. Fixed points were also studied forright-angled Artin groups [17] and lamplighter groups [15].Regarding the continuous extension of an endomorphism to the completion, in-finite fixed points of automorphisms of free groups were also discussed by Bestvinaand Handel in [2] and Gaboriau, Jaeger, Levitt and Lustig in [11]. The dynamicsof free groups automorphisms is proved to be asymptotically periodic in [14]. In [7],the dynamical study of infinite fixed points was performed for monoids defined byspecial confluent rewriting systems (which contain free groups as a particular case).This was also achieved in [19] for virtually injective endomorphisms of virtually freegroups. Endomorphisms of free-abelian times free groups Z m × F n have been studied ∗ [email protected]
1n [9] and their continuous extension to the completion is studied in [5] and the caseof F n × F m with m, n ≥ , k )-quasi-geodesics, which will beseen to arise naturally. We formulate the bounded reduction property in geometricterms throughout Section 4. In Section 5, we show that every nontrivial uniformlycontinuous endomorphism of a hyperbolic group must satisfy a H¨older condition andobserve that these results combined with the work in [16] yield that every endomor-phism of a finitely generated hyperbolic group admitting a continuous extension tothe completion has a finitely generated fixed point subgroup. We now introduce some well-known results on hyperbolic groups. For more details,the reader is referred to [13] and [4].A mapping ϕ : ( X, d ) → ( X ′ , d ′ ) between metric spaces is called an isometricembedding if d ′ ( xϕ, yϕ ) = d ( x, y ), for all x, y ∈ X . A surjective isometric embeddingis an isometry .A metric space ( X, d ) is said to be geodesic if, for all x, y ∈ X , there exists anisometric embedding ξ : [0 , s ] → X such that 0 ξ = x and sξ = y , where [0 , s ] ⊂ R isendowed with the usual metric of R . We call ξ a geodesic of ( X, d ). We shall oftencall Im( ξ ) a geodesic as well. In this second sense, we may use the notation [ x, y ] todenote an arbitrary geodesic connecting x to y . When the endpoint of a geodesic α coincides with the starting point of a geodesic β , we denote the concatenation of bothgeodesics by [ αβ ]. Note that a geodesic metric space is always (path) connected. A quasi-isometric embedding of metric spaces is a mapping ϕ : ( X, d ) → ( X ′ , d ′ ) such2hat there exist constants λ ≥ K ≥ λ d ( x, y ) − K ≤ d ′ ( xϕ, yϕ ) ≤ λd ( x, y ) + K for all x, y ∈ X . We may call it a ( λ, K )-quasi-isometric embedding if we want tostress the constants.If in addition ∀ x ′ ∈ X ∃ x ∈ X : d ′ ( x ′ , xϕ ) ≤ K, we say that ϕ is a quasi-isometry . Two metric spaces ( X, d ) and ( X ′ , d ′ ) are saidto be quasi-isometric if there exists a quasi-isometry ϕ : ( X, d ) → ( X ′ , d ′ ). Quasi-isometry turns out to be an equivalence relation on the class of metric spaces. A( λ, K ) -quasi-geodesic of ( X, d ) is a ( λ, K )-quasi-isometric embedding ξ : [0 , s ] → X such that 0 ξ = x and sξ = y , where [0 , s ] ⊂ R is endowed with the usual metric of R . Given x, y, z ∈ X , a geodesic triangle [[ x, y, z ]] is a collection of three geodesics[ x, y ], [ y, z ] and [ z, x ] in X . Given δ ≥
0, we say that X is δ -hyperbolic if ∀ w ∈ [ x, y ] d ( w, [ y, z ] ∪ [ z, x ]) ≤ δ holds for every geodesic triangle [[ x, y, z ]].Let ( X, d ) be a metric space and
Y, Z nonempty subsets of X . We call the ε -neighbourhood of Y in X and we denote by V ε ( Y ) the set { x ∈ X | d ( x, Y ) ≤ ε } .We call the Hausdorff distance between Y and Z and we denote by Haus( Y, Z ), thenumber defined by inf { ε > | Y ⊆ V ε ( Z ) and Z ⊆ V ε ( Y ) } if it exists. If it doesn’t, we say that Haus( Y, Z ) = ∞ .Given a group H = h A i , consider its Cayley graph Γ A ( H ) with respect to A endowed with the geodesic metric d A , defined by letting d A ( x, y ) to be the length ofthe shortest path in Γ A ( H ) connecting x to y . This is not a geodesic metric space,since d A only takes integral values. However, we can define the geometric realization ¯Γ A ( H ) of its Cayley graph Γ A ( H ) by embedding ( H, d A ) isometrically into it. Then,edges of the Cayley graph become segments of length 1. With the metric inducedby d A , which we will also denote by d A , ¯Γ A ( H ) becomes a geodesic metric space.We say that a group H is hyperbolic if the metric space (¯Γ A ( H ) , d A ) is hyperbolic.We will simply write d instead of d A when no confusion arises. Also, for x ∈ H wewill often denote d A (1 , u ) by | u | .From now on, H will denote a finitely generated hyperbolic group generated bya finite set A and π : e A ∗ → H will be a matched epimorphism.An important property of the class of automatic groups, for which the class ofhyperbolic groups is a subclass, is the fellow traveller property . Given a word u ∈ e A ∗ ,we denote by u [ n ] , the prefix of u with n letters. If n > | u | , then we consider u [ n ] = u .We say that the fellow traveller property holds for L ⊆ e A ∗ if, for every M ∈ N , thereis some N ∈ N such that, for every u, v ∈ L,d A ( uπ, vπ ) ≤ M ⇒ d A ( u [ n ] π, v [ n ] π ) ≤ N, n ∈ N .Given g, h, p ∈ H , we define the Gromov product of g and h taking p as basepointby ( g | h ) Ap = 12 ( d A ( p, g ) + d A ( p, h ) − d A ( g, h )) . We will often write ( g | h ) to denote ( g | h ) A . Notice that, in the free group case, wehave that ( g | h ) = | g ∧ h | . Let G be a subgroup of a hyperbolic group H = h A i and let q ≥
0. We say that G is q - quasiconvex with respect to A if ∀ x ∈ [ g, g ′ ] d A ( x, G ) ≤ q holds for every geodesic [ g, g ′ ] in Γ A ( H ) with endpoints in G . We say that G is quasiconvex if it is q -quasiconvex for some q ≥ Gromov boundary ∂H of H , such as beingthe equivalence classes of geodesic rays, when two rays are considered equivalent ifthe Hausdorff distance between them is finite. Another model for ∂H can be definedusing Gromov sequences . We say that a sequence of points ( x i ) i ∈ N in H is a Gromovsequence if ( x i | x j ) → ∞ as i → ∞ and j → ∞ . Two such sequences ( x i ) i ∈ N and( y j ) j ∈ N are equivalent if lim i,j →∞ ( x i | y j ) = ∞ . The set of all the equivalence classes is another standard model for the boundary ∂H . Identifying an element h in H with the constant sequence ( h ) i , we can extendthe Gromov product to the completion by putting( α | β ) Ap = sup (cid:26) lim inf i,j →∞ ( x i | y j ) Ap (cid:12)(cid:12) ( x i ) i ∈ N ∈ α, ( y j ) j ∈ N ∈ β (cid:27) We define ρ Ap,γ ( g, h ) = (cid:26) e − γ ( g | h ) Ap if g = h p, g, h ∈ H .Given p ∈ H , γ > T ≥
1, we denote by V A ( p, γ, T ) the set of all metrics d on H such that 1 T ρ
Ap,γ ( g, h ) ≤ d ( g, h ) ≤ T ρ
Ap,γ ( g, h ) (1)We refer to the metrics in some V A ( p, γ, T ) as the visual metrics on H . Let d ∈ V A ( p, γ, T ). The metric space ( H, d ) is not complete in general. However,its completion ( b H , ˆ d ) is a compact space and can be obtained by considering b H = H ∪ ∂H . It is a well-known fact that the topology induced by ˆ d on ∂H is the Gromovtopology .Considering the extension of ρ Ap,γ to the boundary, we define, for α, β ∈ ˆ H ˆ ρ Ap,γ ( α, β ) = (cid:26) e − γ ( α | β ) Ap if α = β α, β ∈ ˆ H , the inequalities1 T ˆ ρ Ap,γ ( α, β ) ≤ ˆ d ( α, β ) ≤ T ˆ ρ Ap,γ ( α, β )hold [4, Section III.H.3]. (1 , k ) -quasi-geodesics The goal of this section is to present a slightly more general formulation of thefellow traveller property. While we usually state the property considering geodesics,we can see that we get essentially the same result when the paths considered are(1 , k )-quasi-geodesics for some k ∈ N . We will present the result in a differentversion, considering quasi-geodesics with endpoints at distance at most one fromone another, but we remark that the result holds as long as the distance betweenthe endpoints is bounded by some constant. Proposition 3.1.
Let u be a (1 , r ) -quasi-geodesic and v be a (1 , s ) -quasi-geodesicwith the same starting point. Then, there is a constant N depending on r, s, δ suchthat, for all n ∈ N , we have that d A ( uπ, vπ ) ≤ ⇒ d A ( u [ n ] π, v [ n ] π ) ≤ N. Proof.
Since the Cayley graph of H with respect to A is vertex-transitive, we canassume that 1 is the starting point of both u and v . Let p and q be the endpointsof u and v , and consider a geodesic w from q to p . p q uv w Let k = max { r, s } . Then u, v and w are all (1 , k )-quasi-geodesics. By Corollary 1.8,Chapter III.H in [4], there is a constant δ ′ , depending only on r, s, δ , such that thetriangle ( u, v, w ) is δ ′ -thin. Let n ∈ N . We will prove that d A ( u [ n ] π, v [ n ] π ) ≤ k + 2 δ ′ + 4 . Consider the factorizations of u and v given by1 p n p, u [ n ] u n q n q, v [ n ] v n . Suppose that v n = 1. This means that q n = q , d A (1 , q ) ≤ n and d A (1 , p ) ≤ n + 1 . Also, notice that, since u is a (1 , k )-quasi-geodesic, we have that | u | − k ≤ d A (1 , p ) ≤ | u | . We have that d A ( u [ n ] π, p ) ≤ | u | − n ≤ d A (1 , p ) − n + k ≤ k + 1 ≤ k + 2 δ ′ + 4 . u n = 1 is analogous, so we assume that u n , v n = 1.Since ( u, v, w ) is a δ ′ -thin triangle, there is some m ≤ | u | , such that d A ( v [ n ] π, u [ m ] π ) ≤ δ ′ + 1 . Again, since u and v are (1 , k )-quasi-geodesics, we have that d A (1 , q ) − n − k ≤ | v | − n − k ≤ d A ( q n , q ) ≤ | v | − n ≤ d A (1 , q ) − n + k and d A (1 , p ) − m − k ≤ | u | − m − k ≤ d A ( p m , p ) ≤ | u | − m ≤ d A (1 , p ) − m + k. So, d A (1 , q ) − n − k ≤ d ( q n , q ) ≤ d ( q n , p m ) + d ( p m , p ) + d ( p, q ) ≤ δ ′ + 1 + d A (1 , p ) − m + k + 1 , thus, m − n ≤ δ ′ + 1 + d A (1 , p ) − d A (1 , q ) + 2 k + 1 ≤ δ ′ + 2 k + 3 . Similarly, we havethat d A (1 , p ) − m − k ≤ d ( p m , p ) ≤ d ( p m , q n ) + d ( q n , q ) + d ( q, p ) ≤ δ ′ + 1 + d A (1 , q ) − n + 2 k + 1 , thus, n − m ≤ δ ′ + 1 + d A (1 , q ) − d A (1 , p ) + 2 k + 1 ≤ δ ′ + 2 k + 3 . Hence, we have that d ( p n , q n ) ≤ d ( p n , p m )+ d ( p m , q n ) ≤ | m − n | + k + δ ′ +1 ≤ δ ′ +2 k +3+ k + δ ′ +1 = 3 k +2 δ ′ +4 . In this section, we will present three (equivalent) geometric versions of the BoundedReduction Property (also known as the Bounded Cancellation Lemma) for hyper-bolic groups. The bounded reduction property has proved itself to be a most usefultool in studying the dynamics of (virtually) free group endomorphisms (see, for ex-ample, [2], [8],[11],[18] [19]). We will later use it to prove the main result of thispaper.For free groups, the bounded cancellation lemma is said to hold for an endomor-phism ϕ ∈ End( F n ) if there is some constant N ∈ N such that | u − ∧ v | = 0 ⇒ | u − ϕ ∧ vϕ | < N holds for all u, v ∈ F n , where u ∧ v denotes the longest common prefix between u and v .Since, for u, v ∈ F n , the Gromov product ( u | v ) coincides with u ∧ v , a naturalgeneralization for the hyperbolic case is ∃ N ∈ N (( u | v ) = 0 ⇒ ( uϕ | vϕ ) ≤ N ) .
6n Proposition 15, Chapter 5 in [13], it is shown that if ϕ is a ( λ, K )-quasi-isometric embedding, then there exists a constant A depending on λ, K and δ , withthe property that 1 λ ( u | v ) − A ≤ ( uϕ | vϕ ) ≤ λ ( u | v ) + A. (2)So, if H is a hyperbolic group and ϕ : H → H is a quasi-isometric embedding, thenfor every p ≥
0, there exists q ≥ u | v ) ≤ p ⇒ ( uϕ | vϕ ) ≤ q. We will now present a geometric formulation of the inequality ( u | v ) ≤ p . Lemma 4.1.
Let u, v ∈ H and p ∈ N . Then ( u | v ) ≤ p if and only if for anygeodesics α and β from to u − and v , respectively, we have that the concatenation α / / u − β / / u − v is a (1 , p ) -quasi-geodesic.Proof. Let u, v ∈ H and p ∈ N . Suppose that ( u | v ) ≤ p and take geodesics α and β from 1 to u − and v , respectively. Take the concatenation ζ : [0 , | u | + | v | ] → H ,where 0 ζ = 1, ( | u | ) ζ = u − and ( | u | + | v | ) ζ = u − v. We will prove that ζ is a (1 , p )-quasi-geodesic, i.e., for 0 ≤ i ≤ j ≤ | u | + | v | , wehave that j − i − p ≤ d ( iζ, jζ ) ≤ j − i + 2 p. We can assume that 0 ≤ i ≤ | u | ≤ j ≤ | u | + | v | . Clearly, we have that d ( iζ, jζ ) ≤ j − i ≤ j − i + 2 p .Suppose that | jζ | < j − u | v ) and consider a geodesic γ : [0 , | jζ | ] from 1 to jζ and concatenate it with ζ | [ j, | u | + | v | ] , which gives us a path of length | u | + | v | − j + | jζ | < | u | + | v | − j + j − u | v )= | u | + | v | − u | v )= | u | + | v | − | u | − | v | + | u − v | = | u − v | from 1 to u − v and that contradicts the definition of d . Thus, | jζ | ≥ j − u | v ).So, d ( u − , jζ ) + | u − v | − | v | − d ( iζ, jζ ) − d (1 , iζ ) ≤ d ( u − , jζ ) + | u − v | − | v | − | jζ | = j − | u | + d ( u, v ) − | v | − | jζ | = j − u | v ) − | jζ |≤ , d ( iζ, jζ ) ≥ d ( u − , jζ ) + | u − v | − | v | − d (1 , iζ )= j − | u | + | u − v | − | v | − i = j − i − u | v ) ≥ j − i − p To prove the converse, suppose that, for any geodesics α and β from 1 to u − and v , respectively, we have that the concatenation1 α / / u − β / / u − v is a (1 , p )-quasi-geodesic. Then, take any geodesics α and β in the conditions aboveand consider the (1 , p )-quasi-geodesic ζ obtained by concatenating α and β . Then d ( u, v ) = d (1 , u − v ) = d (0 ζ, ( | u | + | v | ) ζ ) ≥ | u | + | v | − p, so 2( u | v ) = | u | + | v | − d ( u, v ) ≤ p and we are done.Let ϕ : H → H be a map. We say that the BRP holds for ϕ if, for every p ≥ q ≥ u and v such that1 u / / u v / / uv is a (1 , p )-quasi-geodesic, we have that given any two geodesics α , β , from 1 to uϕ and from uϕ to ( uv ) ϕ , respectively, the path1 α / / uϕ β / / ( uv ) ϕ is a (1 , q )-quasi-geodesic. Proposition 4.2.
Let H be a hyperbolic group and ϕ : H → H be a mapping. Then,the BRP holds for ϕ if and only if for every p ≥ there is some q ≥ such that ( u | v ) ≤ p ⇒ ( uϕ | vϕ ) ≤ q. (3) holds.Proof. Let H be a hyperbolic group, ϕ : H → H be a mapping and p be a non-negative integer. Suppose that the BRP holds for ϕ and take u, v ∈ H such that( u | v ) ≤ p . Then, by Lemma 4.1 and the BRP, there is some q ≥ α and β from 1 to u − ϕ and from u − ϕ to ( u − v ) ϕ , respectively, the con-catenation ζ : [0 , | u − ϕ | + | vϕ | ] → H is a (1 , q )-quasi-geodesic. Using Lemma 4.1again, we have that ( uϕ | vϕ ) ≤ q .Now, suppose that for every p ≥
0, there is some q ≥ α , β such that the concatenation1 α / / u β / / uv
8s a (1 , p )-quasi-geodesic. In particular, it is also a (1 , p )-quasi-geodesic. Then byLemma 4.1, we have that ( u − | v ) ≤ p , so, using (3), we have that ( u − ϕ | vϕ ) ≤ q ,so by Lemma 4.1, we have that the path1 α / / uϕ β / / ( uv ) ϕ is a (1 , q )-quasi-geodesic for every geodesics α , β as above.The following proposition is an immediate consequence of (3) and Proposition4.2. Proposition 4.3. If ϕ : H → H is a quasi-isometric embedding, then the BRPholds for ϕ. The next proposition shows that the bounded reduction property can be reducedto the case where p = 0. Proposition 4.4.
Let H be a hyperbolic group and ϕ ∈ End ( H ) . If the BRP holdsfor ϕ for p = 0 , then it holds for every p ∈ N .Proof. Let H be a hyperbolic group and ϕ ∈ End ( H ) and assume that the BRP holdswhen p = 0. Let p ∈ N , u, v ∈ H , and take two geodesics α and β from 1 to u andfrom u to uv , respectively, so that the concatenation [ αβ ] is a (1 , p )-quasi-geodesic.By Proposition 3.1, there is some N ∈ N such that for a (1 , p )-quasi-geodesic ξ starting in 1 and ending in uv , we have that d ( ξ [ n ] , [ αβ ] [ n ] ) < N, (4)for every n ∈ N . We will prove that, there is some q ∈ N such that, given anytwo geodesics ξ and ξ from 1 to uϕ and from uϕ to ( uv ) ϕ, respectively, theirconcatenation is a (1 , q )-quasi-geodesic.Take γ to be a geodesic from 1 to uv and consider the factorization1 γ [ | u | ] / / x γ / / uv. Notice that both γ [ | u | ] and γ are geodesics and their concatenation, γ is also ageodesic. In particular γ is a (1 , p )-quasi-geodesic with the same starting and endingpoints as the concatenation of α and β . So, by, (4) we have that d ( x, u ) < N .Set B ϕ = max {| aϕ | (cid:12)(cid:12) a ∈ e A } . Then, we have that d ( xϕ, uϕ ) ≤ B ϕ d ( x, u ) < B ϕ N. Let ζ and ζ be geodesics from 1 to xϕ and from xϕ to ( uv ) ϕ , respectively. Sincethe BRP holds for ϕ when p = 0, we have that there is some constant p such thatthe concatenation [ ζ ζ ] is a (1 , p )-quasi-geodesic and that constant is independentfrom the choice of γ . So, we have that d (1 , ( uv ) ϕ ) ≥ d (1 , xϕ ) + d ( xϕ, ( uv ) ϕ ) − p . (5)Since d ( uϕ, ( uv ) ϕ ) ≤ d ( uϕ, xϕ ) + d ( xϕ, ( uv ) ϕ ) ≤ N B ϕ + d ( xϕ, ( uv ) ϕ ), we have that d ( xϕ, ( uv ) ϕ ) ≥ d ( uϕ, ( uv ) ϕ ) − N B ϕ . (6)9imilarly, we have that d (1 , uϕ ) ≤ d (1 , xϕ ) + d ( xϕ, uϕ ) ≤ d (1 , xϕ ) + N B ϕ , and so d (1 , xϕ ) ≥ d (1 , uϕ ) − N B ϕ . (7)Combining (5) with (6) and (7), we have that d (1 , ( uv ) ϕ ) ≥ d (1 , uϕ ) − N B ϕ + d ( uϕ, ( uv ) ϕ ) − N B ϕ − p = d (1 , uϕ ) + d (1 , vϕ ) − N B ϕ − p . Hence 2 (cid:0) ( uϕ ) − | vϕ (cid:1) = d (1 , uϕ ) + d (1 , vϕ ) − d (( uϕ ) − , vϕ )= d (1 , uϕ ) + d (1 , vϕ ) − d (1 , ( uv ) ϕ ) ≤ N B ϕ + p . By Lemma 4.1, we have that the concatenation [ ξ ξ ] is a (1 , N B ϕ + p )-quasi-geodesic.If ( X, d ) is δ -hyperbolic and λ ≥ K ≥
0, it follows from [4, Thm 1.7, SectionIII.H.3] that there exists a constant R ( δ, λ, K ), depending only on δ, λ, K , such thatany geodesic and ( λ, K )-quasi-geodesic in X having the same initial and terminalpoints lie at Hausdorff distance ≤ R ( δ, λ, K ) from each other. This constant will beused in the proof of the next result.We recall that for a geodesic α : [0 , n ] → H , we will often denote its image by α as well. We are now ready to present two more (equivalent) geometric formulationsof the BRP. Theorem 4.5.
Let ϕ ∈ End ( H ) . The following conditions are equivalent:(i) the BRP holds for ϕ .(ii) there is some N ∈ N such that, for all x, y ∈ H and every geodesic α = [ x, y ] ,we have that αϕ is at bounded Hausdorff distance to every geodesic [ xϕ, yϕ ] .(iii) there is some N ∈ N such that, for all x, y ∈ H and every geodesic α = [ x, y ] ,we have that αϕ ⊆ V N ( ξ ) for every geodesic ξ = [ xϕ, yϕ ] .Proof. Clearly (ii) ⇒ (iii).(i) ⇒ (ii) : Let x, y ∈ H and N ∈ N given by the BRP when p = 0. Considergeodesics α = [ x, y ] and ξ = [ xϕ, yϕ ]. Let u ∈ ξ and k = d ( xϕ, u ). So, clearly, d ( xϕ, yϕ ) ≥ k . Put B ϕ = max {| aϕ | (cid:12)(cid:12) a ∈ e A } . We may assume that k > B ϕ since,otherwise d ( u, αϕ ) ≤ B ϕ . Since d ( xϕ, xϕ ) = 0 < k and d ( yϕ, xϕ ) ≥ k , there is some n k > d ( xϕ, ( x ( α [ n k ] π )) ϕ ) < k and d ( xϕ, ( x ( α [ n k +1] π )) ϕ ) ≥ k (notice that n k > d ( xϕ, ( x ( α [1] π )) ϕ ) ≤ B ϕ < k ). Consider the following factorization of α x α [ nk ] / / x k α k / / y . Using the BRP, we have that, given geodesics β, γ from xϕ to x k ϕ and from x k ϕ to yϕ , respectively, the concatenation xϕ β / / x k ϕ γ / / yϕ
10s a (1 , N )-quasi-geodesic. Set x k +1 = x ( α [ n k +1] π ). We know that d ( x k ϕ, x k +1 ϕ ) ≤ B ϕ and so k ≤ d ( xϕ, x k +1 ϕ ) ≤ d ( xϕ, x k ϕ ) + B ϕ . Thus, we have that d ( x k ϕ, xϕ ) ≥ k − B ϕ . Let z = [ βγ ] [ k ] . Notice that, since d ( x k ϕ, xϕ ) < k , then z ∈ γ.uxϕ x k ϕ yϕ α k βα [ k ] γ Using the fellow traveller property for (1 , N )-quasi-geodesics, there is some constant M depending only on N and δ such that d ( u, z ) < M . Since γ is a geodesic, then d ( x k ϕ, z ) = k − d ( xϕ, x k ϕ ) ≤ k − ( k − B ϕ ) = B ϕ . Thus, d ( u, x k ϕ ) ≤ d ( u, z ) + d ( z, x k ϕ ) < M + B ϕ and u ∈ V M + B ϕ ( αϕ ). Since u is an arbitrary element of ξ such that d ( u, αϕ ) > B ϕ ,we have that ξ ⊆ V M + B ϕ ( αϕ )Now, let v ∈ α . Since the BRP holds, then, taking any geodesics, β ′ , γ ′ from xϕ to vϕ and from vϕ to yϕ , the concatenation xϕ β ′ / / vϕ γ ′ / / yϕ is a (1 , N )-quasi-geodesic, so Haus([ β ′ γ ′ ] , ξ ) < R ( δ, , N ). In particular, d ( vϕ, ξ ) ≤ R ( δ, , N ). Since v is arbitrary, we have that αϕ ⊆ V R ( δ, ,N ) ( ξ ).Hence Haus( ξ, αϕ ) ≤ max { R ( δ, , N ) , M + B ϕ } .(iii) ⇒ (i) Take N such that (iii) holds and u, v ∈ H . Let α = [1 , u ], β = [ u, uv ] besuch that [ αβ ] is a geodesic and consider γ = [1 , uϕ ] and ζ = [ uϕ, ( uv ) ϕ ]. We wantto prove that there is some M ∈ N such that [ γζ ] is a (1 , M )-quasi-geodesic and thatsuffices by Proposition 4.4. From (iii), we have that αϕ ⊆ V N ( γ ), βϕ ⊆ V N ( ζ ) and[ αβ ] ϕ ⊆ V N ([ γζ ]). Since [ αβ ] is a geodesic, then, given a geodesic ξ = [1 , ( uv ) ϕ ], wehave that ([ αβ ] ϕ ) ⊆ V N ( ξ ) too.1 uϕ ( uv ) ϕ αϕγ ξ ζβϕ Now, we have that there is some x u ∈ ξ such that d ( uϕ, x u ) < N . Suppose that d (1 , x u ) ≥ | γ | and denote ξ [ | γ | ] by y . So, γ and ξ u = ξ [ | x u | ] are two geodesics with thesame starting point that end at bounded distance. By the fellow traveller property,there is some K ∈ N such that d ( uϕ, y )) < K . So, we have that, | ζ | = d ( uϕ, ( uv ) ϕ ) ≤ d ( uϕ, y ) + d ( y, ( uv ) ϕ ) ≤ K + d ( y, ( uv ) ϕ ) . Now, | ξ | = d (1 , y ) + d ( y, ( uv ) ϕ ) = | γ | + d ( y, ( uv ) ϕ ) ≥ | γ | + | ζ | − K. d (1 , x u ) < | γ | , the same inequality can be obtained analogously, considering thegeodesics ζ − and ξ − , since | γ | + | ζ | ≥ | ξ | . So, we have that( u − ϕ | vϕ ) = 12 ( d (1 , uϕ ) + d (1 , vϕ ) − d ( u − ϕ, vϕ ))= 12 ( | γ | + | ζ | − d (1 , ( uv ) ϕ )) = 12 ( | γ | + | ζ | − | ξ | ) ≤ K . By Lemma 4.1, we have that [ γζ ] is a (1 , K )-quasi-geodesic.So, combining Propositions 4.2 and 4.4 with Theorem 4.5, we have proved thefollowing result: Theorem 4.6.
Let ϕ ∈ End ( H ) . The following conditions are equivalent:i. The BRP holds for ϕ .ii. The BRP holds for ϕ when p = 0 .iii. ∀ p > ∃ q > ∀ u, v ∈ H (( u | v ) ≤ p ⇒ ( uϕ | vϕ ) ≤ q ) .iv. ∃ q > ∀ u, v ∈ H (( u | v ) = 0 ⇒ ( uϕ | vϕ ) ≤ q ) .v. there is some N ∈ N such that, for all x, y ∈ H and every geodesic α = [ x, y ] ,we have that αϕ is at bounded Hausdorff distance to every geodesic [ xϕ, yϕ ] .vi. there is some N ∈ N such that, for all x, y ∈ H and every geodesic α = [ x, y ] ,we have that αϕ ⊆ V N ( ξ ) for every geodesic ξ = [ xϕ, yϕ ] . We will start by describing when an endomorphism of a hyperbolic group admits acontinuous extension to the completion. It is well know by a general topology result[10, Section XIV.6] that every uniformly continuous mapping ϕ : H → H ′ admits acontinuous extension to the completion. So, we have the following lemma: Lemma 5.1.
Let ϕ : H → H ′ be a mapping of hyperbolic groups, and let d and d ′ be visual metrics on G and G ′ , respectively. Then, the following conditions areequivalent:1. ϕ is uniformly continuous with respect to d and d ′ ;2. ϕ admits a continuous extension ˆ ϕ : ( b H, ˆ d ) → ( b H ′ , ˆ d ′ ) . A mapping ϕ : ( X, d ) → ( X ′ , d ′ ) between metric spaces satisfies a H¨older condi-tion of exponent r >
K > d ′ ( xϕ, yϕ ) ≤ K ( d ( x, y )) r x, y ∈ X . It clearly implies uniform continuity. We will show that in case ofhyperbolic groups, we have equivalence.In [1], the authors thoroughly study endomorphisms of hyperbolic groups sat-isfying a H¨older condition. In particular, they find several properties equivalent tosatisfying a H¨older condition. An endomorphism ϕ of H is virtually injective if itskernel is finite. Theorem 5.2. [1, Thm 4.3] Let ϕ be a nontrivial endomorphism of a hyperbolicgroup G and let d ∈ V A ( p, γ, T ) be a visual metric on G . Then the following condi-tions are equivalent:(i) ϕ satisfies a H¨older condition with respect to d ;(ii) ϕ admits an extension to b G satisfying a H¨older condition with respect to b d ;(iii) there exist constants P > and Q ∈ R such that P ( gϕ | hϕ ) Ap + Q ≥ ( g | h ) Ap for all g, h ∈ G ;(iv) ϕ is a quasi-isometric embedding of ( G, d A ) into itself;(v) ϕ is virtually injective and Gϕ is a quasiconvex subgroup of G . The authors in [1] conjecture that every uniformly continuous endomorphismsatisfies a H¨older condition. We will give a positive answer to that problem later inthis section.We now present a natural result following from Theorem 4.5.
Corollary 5.3.
Let ϕ ∈ End ( H ) such that the BRP holds for ϕ . Then Hϕ isquasiconvex.Proof. Let x, y ∈ H and take N ∈ N given by condition (ii) of Theorem 4.5. Considergeodesics α = [ x, y ] and ξ = [ xϕ, yϕ ]. We have that Haus( ξ, αϕ ) ≤ N . Let u ∈ ξ .Then d ( u, Hϕ ) ≤ d ( u, αϕ ) ≤ Haus( ξ, αϕ ) ≤ N. Lemma 4.1 in [1] states that a uniformly continuous endomorphism is virtuallyinjective. So, next we will prove that uniform continuity implies the BRP. In thatcase, it follows that uniformly continuous endomorphisms are precisely the onessatisfying a H¨older condition.Let p, q, x, y ∈ H . We have that( x | y ) p = 12 ( d A ( p, x ) + d A ( p, y ) − d A ( x, y )) ≤
12 ( d A ( q, x ) + d A ( q, y ) − d A ( x, y ) + 2 d A ( p, q ))= ( x | y ) q + d A ( p, q )Similarly, ( x | y ) q ≤ ( x | y ) p + d A ( p, q ), so( x | y ) q − d A ( p, q ) ≤ ( x | y ) p ≤ ( x | y ) q + d A ( p, q ) (8)13 roposition 5.4. Let G = h A i and H = h B i be hyperbolic groups and considervisual metrics d ∈ V A ( p, γ, T ) and d ∈ V B ( p ′ , γ ′ , T ′ ) on G and H , respectively.Let ϕ : ( G, d ) → ( H, d ) be an injective uniformly continuous homomorphism.Then, for every M ≥ , there is some N ≥ such that ( u | v ) A ≤ M ⇒ ( uϕ | vϕ ) B ≤ N holds for every u, v ∈ G .Proof. Since ϕ is uniformly continuous, by a general topology result it admits acontinuous extension ˆ ϕ : ( ˆ G, ˆ d ) → ( ˆ H , ˆ d ). Since ˆ ϕ is a continuous map betweencompact spaces, then it is closed, and so it has a closed (thus compact) image.Now, restricting the codomain of ˆ ϕ to the image, we have a continuous bijectionbetween compact spaces, and so it is a homeomorphism. Its inverse, ψ : Im( ˆ ϕ ) → ˆ G is a continuous map between compact spaces, hence uniformly continuous. So, therestriction ψ ′ : (Im( ϕ ) , d ) → ( G, d ) is also uniformly continuous, i.e., ∀ ε > ∃ δ > d ( x, y ) < δ ⇒ d ( xψ ′ , yψ ′ ) < ε ) , which, by construction of ψ ′ , means that ∀ ε > ∃ δ > d ( xϕ, yϕ ) < δ ⇒ d ( x, y ) < ε ) . (9)Using (1), we have that (9) is equivalent to ∀ M ∈ N ∃ N ∈ N (( xϕ | yϕ ) Bp ′ > N ⇒ ( x | y ) Ap > M ) . (10)Since p and p ′ are fixed, we can change the basepoint to 1 using (8). So, (10) becomesequivalent to ∀ M ∈ N ∃ N ∈ N (( x | y ) A ≤ M ⇒ ( xϕ | yϕ ) B ≤ N ) . Proposition 5.5.
Let d ∈ V A ( p, γ, T ) be a visual metric on H and let ϕ be auniformly continuous endomorphism of H with respect to d . Then, the BRP holdsfor ϕ .Proof. If ϕ is injective it follows from Proposition 5.4. Now, in case ϕ is not injective,by Lemma 4.1 in [1], it must have finite kernel K . Consider π : H → H (cid:30) K to bethe projection and the geodesic metric d Aπ on the quotient. Let ϕ ′ : (cid:16) H (cid:30) K, d Aπ (cid:17) → ( H, d A ) be the injective homomorphism induced by ϕ . H HH (cid:30) K ϕπ ϕ ′ Let L = max { d A (1 , x ) | x ∈ K } . g, h ∈ H . We claim that d A ( g, h ) − L ≤ d Aπ ( gπ, hπ ) ≤ d A ( g, h ) . (11)Since h = ga . . . a n implies hπ = ( ga . . . a n ) π for all a , . . . , a n ∈ e A , we have d Aπ ( gπ, hπ ) ≤ d A ( g, h ).Write hπ = ( gw ) π , where w is a word on e A of minimum length. Then h = gwx for some x ∈ K and so d A ( g, h ) ≤ d A ( g, gw ) + d A ( gw, gwx ) = d A (1 , w ) + d A (1 , x ) ≤ | w | + L. By minimality of w , we have actually | w | = d Aπ ( gπ, hπ ) and thus (11) holds.This means that ( H, d A ) and (cid:16) H (cid:30) K, d Aπ (cid:17) are quasi-isometric. In particular, ityields that H (cid:30) K is hyperbolic.Now, take d ′ ∈ V Aπ ( p ′ π, γ ′ , T ′ ) to be a visual metric on H (cid:30) K . For every u, v, p ∈ H, we have that( uπ | vπ ) Aπpπ = 12 ( d Aπ ( pπ, uπ ) + d Aπ ( pπ, vπ ) − d Aπ ( uπ, vπ )) ≤
12 ( d A ( p, u ) + d A ( p, v ) − d A ( u, v ) + L )= ( u | v ) Ap + L ϕ with respect to d , we get that ∀ M ∈ N ∃ N ∈ N (( x | y ) Ap > N ⇒ ( xϕ | yϕ ) Ap > M ) . It follows that ∀ M ∈ N ∃ N ∈ N (( xπ | yπ ) Aπpπ > N ⇒ ( xϕ | yϕ ) Ap > M ) , and so ϕ ′ is uniformly continuous with respect to d ′ and d . It follows from Proposi-tion 5.4 that for every p ≥
0, there is some q ≥ uπ | vπ ) Aπ ≤ p ⇒ ( uπϕ ′ | vπϕ ′ ) A ≤ q (12)holds for all uπ, vπ ∈ H (cid:30) K .Take u, v ∈ H such that ( u | v ) A = 0. Then( uπ | vπ ) Aπ ≤ ( u | v ) A + L L . So, by (12), there is some q which does not depend on u, v such that ( uϕ | vϕ ) A ≤ q . By Proposition 4.4, the BRP holds for ϕ .We can now answer Problem 6.1 left by the authors in [1]. Theorem 5.6.
Let d ∈ V A ( p, γ, T ) be a visual metric on H and let ϕ be an endo-morphism of H . Then ϕ is uniformly continuous with respect to d if and only if theconditions from Theorem 5.2 hold. roof. It is straightforward to see that condition (i) from Theorem 5.2 implies uni-form continuity. Now, if ϕ is uniformly continuous, by Lemma 4.1 in [1] it must bevirtually injective and combining Proposition 5.5 with Corollary 5.3, we have that Hϕ is quasiconvex, so condition (v) of Theorem 5.2 holds.We now present a visual representation of these properties, where the shadedregion represents the nontrivial uniformly continuous endomorphisms. Indeed, wehave proved that every nontrivial uniformly continuous endomorphism satisfies theBRP (Proposition 5.5) and that every endomorphism satisfying the BRP must havequasiconvex image (Corollary 5.3). Also, every virtually injective endomorphismwith quasiconvex image must be uniformly continuous by Theorem 5.2. In [1], theauthors give an example of an injective endomorphism of a torsion-free hyperbolicgroup with non quasiconvex image. So, unlike the case of virtually free groups, theBRP does not hold in general for injective endomorphisms of hyperbolic groups, noteven when restricted to torsion-free hyperbolic groups. In the virtually free groupscase, that does not happen, as every virtually injective endomorphism is uniformlycontinuous [19].Taking ϕ : F → F defined by a a , b b and c
1, we have that F ϕ = h a, b i . Since F ϕ is finitely generated and the standard embedding h a, b i ֒ −→ F is a quasi-isometric embedding, then ϕ has quasiconvex image. But the BRP doesnot hold for ϕ since | cb n ∧ b n | = 0 and | ( cb n ) ϕ ∧ b n ϕ | = | b n ∧ b n | = n , which can bearbitrarily large.Defining an endomorphism with finite image it is easy to find examples of endo-morphisms for which the BRP holds that are not virtually injective, even for virtuallyfree groups. For example, take H = Z × Z and ϕ defined by ( n, (0 ,
0) and( n, (0 , ϕ and its kernel is infinite (in particular,it can’t be uniformly continuous). Quasiconveximage BRP Virtually Injective
Figure 1: Nontrivial uniformly continuous endomorphisms of hyperbolic groupsSo, for hyperbolic groups, we have the figure above in which every region isnonempty. Notice that, for virtually free groups, the only difference is that nontrivialuniformly continuous endomorphisms are precisely the virtually injective ones and16he BRP holds for all of them.In the case of free groups, it is even simpler as, for every nontrivial endomor-phism, the properties of being injective, uniformly continuous and satisfying theBRP are equivalent. Indeed, it is well-known that injective endomorphisms coincidewith uniformly continuous ones and that the BRP holds for this class. It is easyto see that the converse also holds. For n ≥
2, let X = { x , . . . , x n } be a finitealphabet and F n = h X i be a free group of rank n . If a nontrivial endomorphism ϕ ∈ End( F n ) is not injective, then there is some w ∈ Ker( ϕ ) such that w = 1 andsome letter a such that aϕ = 1. Let p = | w | . We have that w is not a proper powerof a since, in that case we would have that wϕ would be a proper power of aϕ andso, nontrivial. So, we have that, for arbitrarily large m ∈ N , | wa m ∧ a m | < | w | , but | ( wa m ) ϕ ∧ a m ϕ | = | a m ϕ | , which is arbitrarily large. So, nontrivial endomorphismsfor which the BRP holds in a free group of finite rank are precisely the injectiveones.In [16], Paulin proved that Fix( ϕ ) is finitely generated if ϕ ∈ Aut( H ). Weremark that its proof also yields the result for quasi-isometric embeddings. So, wehave proved the following result. Theorem 5.7.
Let ϕ ∈ End ( H ) be an endomorphism admitting a continuous ex-tension ˆ ϕ : b H → b H to the completion of H . Then, Fix ( ϕ ) is finitely generated. Acknowledgements
The author is grateful to Armando Martino and to Pedro Silva for fruitful discussionsof these topics, which greatly improved the paper.The author was supported by the grant SFRH/BD/145313/2019 funded byFunda¸c˜ao para a Ciˆencia e a Tecnologia (FCT).
References [1] V. Ara´ujo and P. V. Silva,
H¨older conditions for endomorphisms of hyperbolicgroups , Comm. Algebra, 44(10) (2016), p. 4483-4503.[2] M. Bestvina and M. Handel,
Train tracks and automorphisms of free groups ,Ann. Math. 135 (1992), p. 1-51.[3] O. Bogopolski and O. Maslakova,
An algorithm for finding a basis of the fixedpoint subgroup of an automorphism of a free group , International J. of Algebraand Computation 26(1) (2016), p. 29-67.[4] Martin R. Bridson and Andr´e Haefliger,
Metric spaces of non-positive curvature ,Grundlehren Math. Wissenschaften, Volume 319, Springer, New York, 1999.[5] A. Carvalho,
On the dynamics of extensions of free-abelian times free groupsendomorphisms to the completion , arXiv:2011.05205, preprint, 2020.[6] A. Carvalho,
On endomorphisms of the direct product of two free groups ,arXiv:2012.03635, preprint, 2020. 177] J. Cassaigne and P. V. Silva,
Infinite periodic points of endomorphisms overspecial confluent rewriting systems , Ann. Inst. Fourier 59(2) (2009), p. 769-810.[8] D. Cooper,
Automorphisms of free groups have finitely generated fixed point sets ,J. Algebra 111 (1987), p. 453-456.[9] J. Delgado and E. Ventura,
Algorithmic problems for free-abelian times freegroups , J. Algebra 263(1) (2013), p. 256-283.[10] J. Dugundji,
Topology , Boston, Mass.-London-Sydney: Allyn and Bacon, Inc.Reprinting of the 1966 original, Allyn and Bacon Series in Advanced Mathemat-ics, 1978.[11] D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig,
An index for counting fixedpoints of automorphisms of free groups , Duke Math. J. 93 (1998), p. 425-452.[12] S. M. Gersten,
Fixed points of automorphisms of free groups , Adv. Math. 64(1987), p. 51-85.[13] E. Ghys and P. de la Harpe,
Sur les Groupes Hyperboliques d’apr`es MikhailGromov , Birkh¨auser, Boston, 1990.[14] G. Levitt and M. Lustig,
Automorphisms of free groups have asymptoticallyperiodic dynamics , J. Reine Angew. Math. 619 (2008), p. 1-36.[15] F. Matucci and P. V. Silva,
Extensions of automorphisms of self-similar groups ,J. Group Theory (to appear).[16] F. Paulin,
Points fixes d’automorphismes de groupes hyperboliques , Ann. Inst.Fourier 39 (1989), p. 651-662.[17] E. Rodaro, P. V. Silva and M. Sykiotis,
Fixed points of endomorphisms of graphgroups , J. Group Theory 16(4) (2013), p. 573-583.[18] P. V. Silva,
Fixed points of endomorphisms over special confluent rewritingsystems , Monatsh. Math. 161(4) (2010), p. 417–447.[19] P. V. Silva,