Semi-simple actions of the Higman-Thompson groups T_n on finite-dimensional CAT(0) spaces
aa r X i v : . [ m a t h . G R ] J a n SEMI-SIMPLE ACTIONS OF THE HIGMAN-THOMPSONGROUPS T n ON FINITE-DIMENSIONAL CAT(0) SPACES
MOTOKO KATO
Abstract.
In this paper, we study isometric actions on finite-dimensionalCAT(0) spaces for the Higman-Thompson groups T n , that are general-izations of Thompson’s group T . It is known that all semi-simple ac-tions of T on complete CAT(0) spaces of finite topological dimensionhave global fixed points. After this result, we show that all semi-simpleactions of T n on complete CAT(0) spaces of finite topological dimensionhave global fixed points. In the proof, we regard T n as ring groups ofhomeomorphisms of S introduced by Kim, Koberda and Lodha, anduse general facts on these groups. Introduction
Thompson’s groups F , T and V are finitely generated groups of home-omorphisms of the unit interval, the circle and the Cantor space, respec-tively. These groups are originally studied by Richard Thompson in 1960s,and known to have many interesting properties. In particular, T and V arefirst examples of finitely presented simple groups. there are various general-izations of Thompson’s groups, for example, the Higman-Thompson groups F n , T n and V n . These groups are “ n -adic” versions of F , T and V , where n is a natural number greater than 2 ( F , T and V coincide with F , T and V , respectively). In this paper, we treat actions of these groups on CAT(0)spaces : geodesic spaces whose geodesic triangles are not fatter than the com-parison triangles in the Euclidean plane. The existence of fixed-point freeisometric actions of F , T , V and their generalizations on CAT(0) spaceshave been studied: • T and V have Serre’s property FA, that is, every cellular action onevery simplicial tree has a global fixed point ([7]). • T , V , T n and V n act properly isometrically on CAT(0) cubical com-plexes ([6]). • For T , V , V n and some other generalizations of V , every semi-simpleisometric action on a complete CAT(0) space of finite topologicaldimension has a global fixed point ([10], see also [8]).Based on the last result in [10], we show that every semi-simple isometricaction of T n on a complete CAT(0) space of finite topological dimension hasa global fixed point. In other words, T n has property F A k for semi-simpleactions for every k ∈ N . Basic facts on property F A k can be found in [5]and [13]. The author is supported by JSPS KAKENHI Grant-in-Aid for Research Activity Start-up, Grant Number 19K23406 and JSPS KAKENHI Grant-in-Aid for Young Scientists,Grant Number 20K14311.
In the proof of the main result, we treat T n as groups generated by orien-tation preserving homeomorphisms of S whose supports form a “ring” in S . Such groups were considered by Kim, Koberda and Lodha (cf. [11]) andcalled ring groups of homeomorphisms of S . The family of ring groups arebroad, containing uncountably many isomorphism classes of groups ([11]).In Section 2, we recall precise definitions of F , T , F n and T n and ringgroups. We construct a new generating sets for T n and regard them as ringgroups. In Section 3, we show that for every m -ring group G with m ≥ H of the commutator subgroup of G ,every semi-simple isometric action on a CAT(0) space of finite topologicaldimension has a point fixed by all elements of H (Theorems 3.4). Theproof is based on general facts on ring groups ([11]), and techniques to treatisometric actions of T on CAT(0) spaces ([10]). As a corollary, it follows thatevery semi-simple isometric action of T n with n ≥ Thompson’s groups, chain groups and ring groups
Convention 2.1.
In the following, we fix an orientation on S . We definean open interval in S as a subset of the form ( a, b ) = { t ∈ S | a < t < b } ,where a = b ∈ S and < is the cyclic order on S . For an open interval J = ( a, b ) in S , we write a = ∂ − J and b = ∂ + J . We also fix a basepointon S , and identify S with [0 , / { } so that 0 = 1 corresponds to thebasepoint. Through this identification, we regard subintervals of [0 ,
1] assubsets of S . We also regard homeomorphisms of [0 ,
1] as homeomorphismsof S that fix the basepoint. For f ∈ Homeo( S ), we write supp( f ) forthe set-theoretic support of f : supp( f ) = { t ∈ S | f ( t ) = t } . For S ⊂ Homeo( S ), we write supp( S ) for the union of supp( s ) for all s ∈ S . We notethat supp( f ) and supp( S ) are open subsets of S . For f , g ∈ Homeo( S ),we write g f for the conjugate gf g − .We start with definitions of Thompson’s groups F and T , following [4]. F is a group of piecewise affine homeomorphisms of the unit interval [0 , standard dyadic interval is a subinterval of [0 ,
1] of theform [ a/ b , ( a + 1) / b ], where a , b are nonnegative integers. T is a group ofpiecewise affine homeomorphisms of the circle S , differentiable with deriva-tives of powers of 2 on finitely many standard dyadic intervals. We identifydivisions of [0 ,
1] into finitely many standard dyadic intervals with finiterooted binary trees, and represent elements of Thompson’s groups by a pairof finite rooted binary trees with the same number of leaves (with an indi-cation of the basepoint when we consider T ). The next lemma shows basicfacts on Thompson’s groups. Lemma 2.1. (cf. [4])(1) The commutator subgroup [
F, F ] of F is simple, and coincides withthe subgroup consisting of elements which restrict to the identity onsome neighborhoods of 0 and 1.(2) Let G be F or T . For every standard dyadic interval I , let G ( I ) = { g ∈ G | supp( g ) ⊂ I } < G . Then G ( I ) is isomorphic to F , and CTIONS OF THE HIGMAN-THOMPSON GROUPS ON CAT(0) SPACES 3 its commutator subgroup [ G ( I ) , G ( I )] coincides with the subgroupconsisting of elements whose supports are subsets of the interior of I .(3) Let I be a standard dyadic interval. For every nonempty open subset J of the interior of I and every compact subset J of the interiorof I , there exists f in the commutator subgroup of F ( I ) such that f ( J ) is contained in J .Lemma 2.1 (1) is Theorem 4.1 of [4]. (2) follows from (1) through theidentification of [0 ,
1] and a standard dyadic interval. (3) is immediate from(1) and the definition of F .As a generalization of F and T , we can consider n -adic versions of F and T . For every natural number n ≥
2, we define standard n -adic intervals as intervals of the form [ a/n b , ( a + 1) /n b ] in [0 , ( n -adic) Higman-Thompson groups F n (resp. T n ) is a group of piecewise affine homeomor-phisms of the unit interval [0 ,
1] (resp. S ), differentiable with derivativesof powers of n on finitely many standard n -adic intervals (cf. [3], [9]). Bydefinition, F and T coincides with F and T . We identify n -adic divisionsof [0 ,
1] with finite rooted n -ary trees and represent elements of the n -adicHigman-Thompson groups by a pair of finite rooted n -ary trees with thesame number of leaves. Figure 1 shows a 2-caret, a 3-caret and an n -caret. nn -caret3-caret2-caret Figure 1. n -carets the i -th leaf n n nx n,i (1 ≤ i ≤ n − n n n nx n,n n nn n ny n basepoint0 ∈ [0 ,
1] basepoint1 − /n ∈ [0 , Figure 2.
Generators of F n and T n Lemma 2.2 ([3, Section 4C]) . For every natural number n ≥ F n isgenerated by x n,i (1 ≤ i ≤ n ), where x n,i ( t ) = t (0 ≤ t ≤ i − n ) n ( t − i − n ) + i − n ( i − n ≤ t ≤ in − in ) t − ( in + in ) + n − n ( in − in ≤ t ≤ in ) n ( t −
1) + 1 ( in ≤ t ≤ , M. KATO J J J J J J J J S Figure 3.
A chain and a ring of intervalsfor 1 ≤ i ≤ n −
1, and x n,n ( t ) = t (0 ≤ t ≤ − n ) n ( t − (1 − n )) + 1 − n (1 − n ≤ t ≤ − n + n − n ) t + n − n + n (1 − n + n − n ≤ t ≤ − n + n ) n ( t −
1) + 1 (1 − n + n ≤ t ≤ .T n is generated by F n and y n , where y n ( t ) = ( t + 1 − n (0 ≤ t ≤ n ) t − n ( n ≤ t ≤ . To make calculations easier, we represent generators of the Higman-Thompson groups as tree pairs in Figure 2. In this figure, each trianglewith sign n denotes an n -caret.We also treat another generalization of F , defined by Kim, Koberda andLodha [11]. Let m ≥ m -chain is an ordered n -tuple of open intervals ( J , . . . , J m ) in R or S , satisfying that • J i − J i +1 and J i +1 − J i are not empty for every 1 ≤ i ≤ m −
1, and • J i ∩ J j = ∅ if | i − j | ≥ m -prechain group G F is a subgroup of Homeo + ([0 , F = { f i } ≤ i ≤ n ⊂ Homeo + ([0 , f ) , . . . , supp( f m )) is an m -chain and that S i supp( f i ) =(0 , m -chain group is an m -prechain group such that h f i , f i +1 i is iso-morphic to F for every 1 ≤ i ≤ m −
1. A group is a chain group if it isan m -chain group for some m . A chain subgroup of a chain group G F is asubgroup generated by elements of F with consecutive indices. The familyof chain groups is broad, in fact, there exist uncountably many isomorphismtypes of 3-chain groups (Theorem 1.7 of [11]).It is known that F m is a “stabilization” of m -chain groups. Lemma 2.3 ([11, Proposition 1.10]) . Let G F be an m -prechain group withrespect to the generating set { f i } ≤ i ≤ m , where m ≥
2. For all but finitelymany N ∈ N , the subgroup h{ f Ni } ≤ i ≤ m i of G F is a chain group. Moreover,for all but finitely many N ∈ N , h{ f Ni } ≤ i ≤ m i is isomorphic to F m .Let m ≥ m -ring is an ordered m -tuple of openintervals ( J , . . . , J m ) in S , satisfying that • J i − J i +1 and J i +1 − J i are not empty for every i and • J i ∩ J j = ∅ if 2 ≤ | i − j | ≤ m − CTIONS OF THE HIGMAN-THOMPSON GROUPS ON CAT(0) SPACES 5 where indices are considered by modulo m . An m -ring group G F is a sub-group of Homeo + ( S ) generated by F = { f i } ≤ i ≤ m ⊂ Homeo + ( S ) suchthat (supp( f ) , . . . , supp( f m )) is an m -ring, where h f i , f i +1 i is isomorphic to F for every i modulo m . Figure 3 shows an example of a 4-ring of intervals.A group is called a ring group if it is an m -ring group for some m . Chainsubgroups of an m -ring group is defined similarly as chain subgroups of chaingroups, considering indices of generators by modulo m .The followings are useful facts on chain and ring groups. Lemma 2.4 ([11, Lemma 3.1]) . Let f , g ∈ Homeo + ( R ) be elements whosesupports are open intervals of the form ( a, b ) and ( a ′ , b ′ ) respectively, where a < a ′ < b < b ′ . If gf ( a ′ ) ≥ b , then h f, g i ∼ = F . Lemma 2.5 ([11, Lemma 3.6]) . Let G F be an m -prechain group for some m ≥
2. Let t ∈ (0 ,
1) be a boundary point of supp( f ) of some f ∈ F .For every closed interval I ⊂ (0 ,
1) and every open neighborhood J of t in(0 , g in the commutator subroup of G F such that g ( I ) ⊂ J . Lemma 2.6 (cf. [11, Proposition 1.5]) . Let G F be an m -ring group for some m ≥
4. Then for all m ′ ≥ m , there exists an m ′ -ring group H such that G ∼ = H . Proof.
Although the proof is similar as the corresponding result for chaingroups ([11, Proposition 1.5]), we add some arguments needed for ringgroups. It is enough to show the case where m ′ = m + 1.First, we give a new generating set for G F as an ( m + 1)-ring group. Let F = { f i } ≤ i ≤ m . Let f ′ m − = f − Nm − f m − , f ′ m = ( f m − f m ) N f Nm − , f ′ m − = ( f ′ m ) − f m − , f ′ m +1 = f m for some N ∈ N . We further define f ′ i (1 ≤ i ≤ m −
3) by f ′ i = ( f Nm f ( i = 1) f i (otherwise) , and let F ′ = { f ′ i } ≤ i ≤ m +1 . By the construction of F ′ , G F ′ = G F . If N issufficiently large, (supp( f ′ i )) ≤ i ≤ m +1 is a chain of intervals and h f ′ i , f ′ i +1 i = F for 2 ≤ i ≤ m ([11, Theorem 4.7]).Next, we confirm that (supp( f ′ i )) i = m,m +1 , , is a chain of intervals. In thefollowing, we write < for the induced order on some open interval in S .Since f ′ m = ( f m − f m ) N f m − and f ′ = ( f m ) N f , ∂ + supp( f ′ m ) = ( f m − f m ) N ( ∂ + supp( f Nm − )) = ( f m − f m ) N ( ∂ + supp( f m − ))= f Nm ( ∂ + supp( f m − )) < f Nm ( ∂ − supp( f )) = ∂ − supp( f ′ ) . Here, the third equality follows from an observation that f im ( ∂ + supp( f m − )) ∈ supp( f m ) for all i ∈ Z and f m − ( x ) = x for all x ∈ supp( f m ). More-over, since f m fixes supp( f m ), both ∂ + supp( f ′ m ) and ∂ − supp( f ′ ) are insupp( f m ) = supp( f ′ m +1 ). Therefore, ∂ − supp( f ′ m +1 ) < ∂ + supp( f ′ m ) < ∂ − supp( f ′ ) < ∂ + supp( f ′ m +1 ) . M. KATO
Since f ′ is either f ( m ≥
5) or f − N f ( m = 4), we have ∂ − supp( f ′ ) = ∂ − supp( f ) in either case. Therefore, ∂ + supp( f ′ m +1 ) < ∂ − supp( f ′ ) < ∂ + supp( f ′ ) . It follows that (supp( f ′ i )) i = m,m +1 , , is a chain of intervals.Finally, we show that h f ′ i , f ′ i +1 i = F for i = m + 1 and 1. For i = m + 1, h f ′ m +1 , f ′ i = h f m , f mM f i = h f mM f m , f mM f i = h f m , f i = F. Similarly, for i = 1, h f ′ , f ′ i = ( h f mM f , f ′ i = h f , f ′ i = h f , f i = F ( m ≥ h f ′ , f ′ i = h f , f ′ i = h f , f − N f i = h f , f i = F ( m = 4) . Therefore, h f ′ i , f ′ i +1 i = F for i = m − m + 1 and 1. (cid:3) Proposition 2.7.
For every n ≥ T n is an ( n + 1)-ring group. Proof.
First, we find generators of T n whose supports form a ring of intervals.For every natural number n ≥
2, we define g n, and g n, ∈ F n by g n, ( t ) = n t (0 ≤ t ≤ − n ) n ( t − (1 − n )) + n − n (1 − n ≤ t ≤ − n + n ) t − (1 − n ) (1 − n + n ≤ t ≤ − n ) n ( t − (1 − n )) + n − n (1 − n ≤ t ≤ − n + n ) n ( t −
1) + 1 (1 − n + n ≤ t ≤ , and g n, ( t ) = n t (0 ≤ t ≤ − n ) t − n − n (1 − n ≤ t ≤ − n + n ) n ( t − (1 − n + n )) + n (1 − n + n ≤ t ≤ − n ) n ( t − (1 − n )) + 1 − n (1 − n ≤ t ≤ − n + n ) n ( t − (1 − n + n )) + 1 − n (1 − n + n ≤ t ≤ − n ) t (1 − n ≤ t ≤ . Figure 5 describes g n, and g n, as tree pairs. Let f n,i = x − n,i +1 x n,i (1 ≤ i ≤ n − f n,n = x n,n , and f n,n +1 = g n, ( y n g n, ). Figure 4 describes f n,i (1 ≤ i ≤ n + 1) as tree pairs. Since F n is generated by { x n,i } ≤ i ≤ n , elements f n,i = x − i +1 x i (1 ≤ i ≤ n ) generate F n . Since T n is generated by F n and y n that can be written as a product of elements of F n and f n,n +1 , elements f n,i (1 ≤ i ≤ n + 1) generate T n . Here, the supports of these generators are:supp( f n,i ) = (cid:18) i − n , i + 1 n (cid:19) (1 ≤ i ≤ n − , supp( f n,n − ) = (cid:18) − n , − n + 1 n (cid:19) , supp( f n,n ) = (cid:18) − n , (cid:19) , supp( f n,n +1 ) = (cid:18) , n (cid:19) ∪ (cid:18) − n , (cid:19) . Therefore, supp( f n,i ) (1 ≤ i ≤ n + 1) form a ring of open intervals. CTIONS OF THE HIGMAN-THOMPSON GROUPS ON CAT(0) SPACES 7 nn nn the i -th leaf the ( i + 1)-th leaf f n,i (1 ≤ i ≤ n − n nn the ( n − n nnf n,n − nn n n n nn nf n,n +1 n nn n n nf n,n basepoint basepoint Figure 4.
New generators of T n Next, we show that h f n,i , f n,i +1 i (1 ≤ i ≤ n ) and h f n,n +1 , f n, i are iso-morphic to F , by using Lemma 2.4. For every n ≥ ≤ i ≤ n −
3, wehave f n,i +1 f n,i (cid:18) i + 1 n (cid:19) = i + 2 n − n ≥ i + 1 n ,f n,n − f n,n − (cid:18) − n (cid:19) = 1 − n + 1 n − n ≥ − n ,f n,n f n,n − (cid:18) − n (cid:19) ≥ − n ≥ − n + 1 n . By Lemma 2.4, h f n,i , f n,i +1 i (1 ≤ i ≤ n −
1) are isomorphic to F . To dealwith the remaining cases, we consider the conjugate of generators by the1 /n rotation y − n . Thensupp( y − n f n,n ) = (cid:18) , n (cid:19) , supp( y − n f n,n +1 ) = (cid:18) n − n , n (cid:19) , supp( y − n f n, ) = (cid:18) n , n (cid:19) , and we have( y − n f n,n +1 )( y − n f n,n ) (cid:18) n − n (cid:19) = y − n (cid:18) − n + 1 n (cid:19) ≥ y − n (0) , ( y − n f n, )( y − n f n,n +1 ) (cid:18) n (cid:19) = y n − (cid:18) n − n − n (cid:19) ≥ y n − (cid:18) n (cid:19) . Applying Lemma 2.4, h y − n f n,n , y − n f n,n +1 i and h y − n f n,n +1 , y − n f n, i , and thus h f n,n , f n,n +1 i and h f n,n +1 , f n, i are isomorphic to F . It follows that every T n has the structure of an ( n + 1)-ring group. (cid:3) Fixed point properties
In this section, we argue the existence of fixed points for isometric groupactions on CAT(0) spaces of finite topological dimension. In the following,we say a space is k -dimensional if the topological dimension of the space is M. KATO n n n nnng n, n n n nn ng n, the ( n − Figure 5.
Elements g n, and g n, of F n k . We assume that all group actions are isometric, and that all spaces arecomplete. We say that an isometry γ of a metric space ( X, d ) is semi-simple if there exists x ∈ X such that d ( x, γ ( x )) = inf { d ( y, γ ( y )) | y ∈ X } . Anisometric group action is called semi-simple if every group element acts asa semi-simple isometry.To study group actions on finite-dimensional CAT(0) spaces, we use thefollowing two theorems. Theorem 3.1 ([1, Proposition 3.4]) . Let X be a k -dimensional CAT(0)space. Let k , . . . , k l ∈ N such that 0 < k < Σ li =1 k i . Let { S i } ≤ i ≤ l bemutually commuting conjugates in Isom( X ). If each k i -element subset of S i has a fixed point in X for all i ∈ { , ..., l } , then for every i ∈ { , ..., l } everyfinite subset of S i has a fixed point.Theorem 3.1 is a variation of Helly’s Theorem for convex subsets in Eu-clidean spaces. Theorem 3.2 ([10, Theorem 1.1]) . Let k ∈ N . Let G be a group actingfaithfully on a set A . Let g ∈ G be an element satisfying the followingconditions: there exists a sequence of subgroups h g i = H < H < · · · For every f ∈ [ F, F ] and every semi-simple action of F on afinite-dimensional CAT(0) space, f has a fixed point. Proof. We fix k ∈ N arbitrarily. We note that F is acting faithfully on[0 , f is an element of the commutator subgroup [ F, F ], supp( f ) iscontained in a closed interval I in (0 , 1) (Lemma 2.1 (1)). Let { I i } ≤ i ≤ k bea sequence of standard dyadic intervals in (0 , 1) such that each I i is con-tained in the interior of I i +1 . By Lemma 2.1 (3), there exists g ∈ F thatmaps I in the interior of I . Here, supp( g f ) = g supp( f ) is contained in theinterior of I . For every 1 ≤ i ≤ k , let H i be the commutator subgroup of F ( I i ) = { h ∈ F | supp( h ) ⊂ I i } < F , and let H k +1 = F . By Lemma 2.1(2), every H i is isomorphic to the commutator subgroup of F and thusnonabelian and simple. It follows that { H i } ≤ i ≤ k satisfies the first assump-tion in Theorem 3.2. By Lemma 2.1 (3), there exist elements { g i } ≤ i ≤ k such that g i ∈ H i +1 and g i ( I i ) ∩ I i = ∅ . By construction, { H i } ≤ i ≤ k +1CTIONS OF THE HIGMAN-THOMPSON GROUPS ON CAT(0) SPACES 9 supp( f )supp( f ) supp( f )supp( f ) I j [ H , H ] [ H , H ]1 234 1 234 Figure 6. A construction of g j and { g i } ≤ i ≤ k satisfies the second assumption in Theorem 3.2. Therefore,according to Theorem 3.2, for every semi-simple action of F on every k -dimensional CAT(0) space, g f has a fixed point. Since g f is a conjugate of f , f also has a fixed point. (cid:3) Theorem 3.4. Let m ∈ N ≥ . Let G F be an m -ring group with respect tothe generating set F = { f i } ≤ i ≤ m . Let H be a finitely generated subgroupof the commutator subgroup [ G F , G F ]. Then for every semi-simple actionof G F on a finite-dimensional CAT(0) space, elements of H have a commonfixed point. Proof. We fix k ∈ N and a semi-simple action of G F on a k -dimensionalCAT(0) space X arbitrarily. It is enough to show that all elements of H fixa common point of X .First, we construct a finite generating set of H consisting of elementswith fixed points in X . Let c i = [ f i , f i +1 ] and H i = h f i , f i +1 i for 1 ≤ i ≤ m ,where indices are considered modulo m . Since [ G F , G F ] is generated byconjugates of { c i } ≤ i ≤ m , there exists a finite generating set S = { c ′ j } ≤ j ≤ m ′ of H consisting of conjugates of c i . Applying Lemma 3.3 to the inducedaction of H i ∼ = F on X , we see that c i and c ′ j fix a point in X .Next, we replace H with a finitely generated H ′ such that H < H ′ < [ G F , G F ], generated by elements of small supports with fixed points. Sinceevery supp( c i ) is contained in a closed interval [ ∂ − supp( f i ) , ∂ + supp( f i +1 )]in S , there exists a closed interval I j in S such that supp( c ′ j ) ⊂ I j for every j . According to Lemma 2.5, for every j , there exists g j ∈ [ G F , G F ] suchthat g j = g ′ j, · · · g ′ j,l j ( g ′ j,i j ∈ S ≤ i ≤ m [ H i , H i ]) and g j ( I j ) ⊂ supp( H n ). Withsuch a g j , supp( g j c ′ j ) = g j (supp( c ′ j )) is contained in supp( H n ). We showan example of the construction of g j in Figure 6. Let S ′ = { g j c ′ j } ≤ j ≤ m ∪{ g ′ j,i j } ≤ i j ≤ l j , ≤ j ≤ m ′ . Then H is a subgroup of H ′ = h S ′ i and every s ′ ∈ S ′ fixes a point in S ′ , since s ′ ∈ S ′ is a conjugate of an element of [ H i , H i ]. Wenote that for every s ′ ∈ S ′ , there exists i such that supp( s ′ ) ⊂ supp( H i ).Finally, we show that elements of S ′ have a common fixed point in X .According to Lemma 2.6, we may assume that m >> k , so that the supportof every ( k + 1)-element subset S ′ k of S ′ is included in a closed interval inthe support of a chain subgroup L of G F . By applying Lemma 2.5 to L , wecan take ( k + 1) elements of G F which map supp( S ′ k +1 ) into ( k + 1) disjointopen subsets of supp( L ). By applying Theorem 3.1 to S ′ , it follows thatelements of S ′ have a common fixed point. (cid:3) Corollary 3.5. For every n ≥ 3, every semi-simple isometric action of T n on a complete CAT(0) space of finite topological dimension has a globalfixed point. Proof. Let T n be acting semi-simply on a finite-dimensional complete CAT(0)space X . For every n ≥ 3, [ T n , T n ] is a finite index subgroup of T n ([3]), andthus [ T n , T n ] is finitely generated. Applying Theorem 3.4 for G F = T n and H = [ T n , T n ], we see that the set Y of common fixed points of [ T n , T n ] in X is nonempty. Since Y is a closed convex subset of X (cf. [2, PropositionII-6.2]), Y is a finite-dimensional CAT(0) space. Therefore, for the inducedaction of the finite group T n / [ T n , T n ] on Y , there exists a global fixed point y ∈ Y (cf. [2, Proposition II-2.8]). Then y ∈ X is fixed by every element of T n . (cid:3) Remark 3.1. For a chain group G F and a finitely generated subgroup H of [ G F , G F ], we can also get a similar result as Theorem 3.4. In fact, if G F is acting semi-simply on a k -dimensional CAT(0)space X , we can constructa generating set S of H as in the proof of Theorem 3.4. By construction, forevery ( k + 1)-element subset S k of S , S s ∈ S k supp( s ) is included in a closedinterval in (0 , k +1 elements of G F which mapsupp( S k +1 ) into ( k + 1) disjoint open subsets of supp( L ). By Theorem 3.1,it follows that all elements of S and thus H has a common fixed point in X .However, unlike ring groups, a chain group G F never admits Serre’s prop-erty FA. Property FA for a group G is equivalent to the three conditions: G is finitely generated, G does not surject on Z , and G is not an amalgamatedproduct ([12]). We fix a map w : G F → ∪ l ∈ N ( F ∪ F − ) l , which maps eachelement g ∈ G F to a finite word in F ∪ F − = { f ± i } ≤ i ≤ m which represents g . Let p ( g ) = (the number of f in w ( g )) − (the number of f − in w ( g )).By observing the action of g on a neighborhood of 0 ∈ [0 , p ( g ) does not depend on the choice of w . It follows that p : G F → Z is a well-defined surjection, and thus G F does not have property FA. References [1] M. R. Bridson, On the dimension of CAT(0) spaces where mapping class groups act ,J. Reine Angew. Math. , Springer-Verlag, Berlin, 1999.[3] K. S. Brown, Finiteness properties of groups , J. Pure Appl. Algebra, (1-3), 45–75,1987.[4] J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thomp-son’s groups , Enseign. Math. (2) , 215–256, 1996.[5] B. Farb, Group actions and Helly’s theorem , Advances in Mathematics (2009)1574–1588.[6] D. S. Farley, Actions of picture groups on CAT(0) cubical complexes , Geom. Dedicata , 221–242, 2005.[7] D. S. Farley, A proof that Thompson’s groups have infnitely many relative ends , J.Group Theory , 649–656, 2011.[8] A. Genevois, Hyperbolic and cubical rigidities of Thompson’s group V , J. Group The-ory Finitely presented infinite simple groups , Notes Pure Math. , AustralianNational University, Canberra, vii+82 pp, 1974. CTIONS OF THE HIGMAN-THOMPSON GROUPS ON CAT(0) SPACES 11 [10] M. Kato, On groups whose actions on finite-dimensional CAT(0) spaces have globalfixed points , preprint, arXiv:math/1804.10506.[11] S. Kim, T. Koberda and Y. Lodha, Chain groups of homeomorphisms of the intervaland the circle , Ann. Sci. de l’ENS (4) , 797–820, 2019.[12] J.-P. Serre, Trees , Springer-Verlag, Berlin, 2003.[13] O. Varghese, Fixed points for actions of Aut(Fn) on CAT(0) spaces , M¨unster J. ofMath. , 439–462, 2014.(M. Kato) Department of Mathematics, Faculty of Science, Ehime Univer-sity, 2-5 Bunkyo-cho, Matsuyama, Ehime, 790-8577 Japan Email address ::