On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups
aa r X i v : . [ m a t h . G R ] A p r ON EQUIVARIANT HOMEOMORPHISMS OFBOUNDARIES OF CAT(0) GROUPSAND COXETER GROUPS
TETSUYA HOSAKA
Abstract.
In this paper, we investigate an equivariant homeomor-phism of the boundaries ∂X and ∂Y of two proper CAT(0) spaces X and Y on which a CAT(0) group G acts geometrically. We provide a suf-ficient condition and an equivalent condition to obtain a G -equivarianthomeomorphism of the boundaries ∂X and ∂Y as a continuous extensionof the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy , where x ∈ X and y ∈ Y . In this paper, we say that a CAT(0) group G is equivariant (boundary) rigid , if G determines its ideal boundary by theequivariant homeomorphisms as above. As an application, we introducesome examples of (non-)equivariant rigid CAT(0) groups and we showthat if Coxeter groups W and W are equivariant rigid as reflectiongroups, then so is W ∗ W . We also provide a conjecture on non-rigidityof boundaries of some CAT(0) groups. Introduction
In this paper, we investigate an equivariant homeomorphism of the bound-aries of two proper CAT(0) spaces on which a CAT(0) group acts geomet-rically as a continuous extension of a quasi-isometry of the two CAT(0)spaces.Definitions and details of CAT(0) spaces and their boundaries are foundin [8] and [23]. A geometric action on a CAT(0) space is an action byisometries which is proper ([8, p.131]) and cocompact. We note that everyCAT(0) space on which some group acts geometrically is a proper space ([8,p.132]). A group G is called a CAT(0) group , if G acts geometrically onsome CAT(0) space X .It is well-known that if a Gromov hyperbolic group G acts geometricallyon a negatively curved space X , then the natural map G → X ( g gx )extends continuously to an equivariant homeomorphism of the boundariesof G and X . Also if a Gromov hyperbolic group G acts geometricallyon negatively curved spaces X and Y , then the boundaries of X and Y Date : March 30, 2014.2000
Mathematics Subject Classification.
Key words and phrases.
CAT(0) space; CAT(0) group; boundary; geometric action;equivariant homeomorphism; Coxeter group.Partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Edu-cation, Culture, Sports, Science and Technology, Japan. (No. 25800039). re G -equivariant homeomorphic. Indeed the natural map Gx → Gy ( gx gy ) extends continuously to a G -equivariant homeomorphism ofthe boundaries of X and Y . The boundaries of Gromov hyperbolic groupsare quasi-isometric invariant (cf. [8], [12], [23], [24], [25]).Here in [25], Gromov asked whether the boundaries of two CAT(0) spaces X and Y are G -equivariant homeomorphic whenever a CAT(0) group G acts geometrically on the two CAT(0) spaces X and Y . In [7], P. L. Bowersand K. Ruane have constructed an example that the natural quasi-isometry Gx → Gy ( gx gy ) does not extend continuously to any map betweenthe boundaries ∂X and ∂Y of X and Y . Also S. Yamagata [52] has con-structed a similar example using a right-angled Coxeter group and its Daviscomplex. Moreover, there is a research by C. Croke and B. Kleiner [14] onan equivariant homeomorphism of the boundaries ∂X and ∂Y .Also, C. Croke and B. Kleiner [13] have constructed a CAT(0) group G which acts geometrically on two CAT(0) spaces X and Y whose boundariesare not homeomorphic, and J. Wilson [51] has proved that this CAT(0)group has uncountably many boundaries. Recently, C. Mooney [43] hasshowed that the knot group G of any connected sum of two non-trivial torusknots has uncountably many CAT(0) boundaries.Also, it has been observed by M. Bestvina [5] that all the boundaries ofa given CAT(0) group are shape equivalent and R. Geoghegan and P. On-taneda have proved this in [22]. Bestvina has asked the question whetherall the boundaries of a given CAT(0) group are cell-like equivalent. Thisquestion is an open problem and there are some resent research (cf. [44]).The purpose of this paper is to provide a sufficient condition and anequivalent condition to obtain a G -equivariant homeomorphism between thetwo boundaries ∂X and ∂Y of two CAT(0) spaces X and Y on which aCAT(0) group G acts geometrically as a continuous extension of the naturalquasi-isometry Gx → Gy ( gx gy ), where x ∈ X and y ∈ Y .Now we recall the example of Bowers and Ruane in [7]. Let G = F × Z and X = Y = T × R , where F is the rank 2 free group generated by { a, b } and T is the Cayley graph of F with respect to the generating set { a, b } .Then we define the action “ · ” of the group G on the CAT(0) space X by( a, · ( t, r ) = ( a · t, r ) , ( b, · ( t, r ) = ( b · t, r ) , (1 , · ( t, r ) = ( t, r + 1) , for each ( t, r ) ∈ T × R = X , and also define the action “ ∗ ” of the group G on the CAT(0) space Y by( a, ∗ ( t, r ) = ( a · t, r ) , ( b, ∗ ( t, r ) = ( b · t, r + 2) , (1 , ∗ ( t, r ) = ( t, r + 1) , or each ( t, r ) ∈ T × R = Y . Then the group G acts geometrically on thetwo CAT(0) spaces X and Y , and the quasi-isometry g · x g ∗ y (where x = (1 , ∈ X and y = (1 , ∈ Y ) does not extend continuously to anymap from ∂X to ∂Y . Indeed for g i = a i b i ∈ F , { g ∞ i | i ∈ N } → a ∞ as i → ∞ in ∂T , lim n →∞ ( g ni , · x = [ g ∞ i , , lim n →∞ ( a n , · x = [ a ∞ , , in X ∪ ∂X , and lim n →∞ ( g ni , ∗ y = [ g ∞ i , π , lim n →∞ ( a n , ∗ y = [ a ∞ , , in Y ∪ ∂Y . Hence any map from ∂X to ∂Y obtained as a continuouslyextension of the quasi-isometry G · x → G ∗ y ( g · x → g ∗ y ) mustsend [ g ∞ i ,
0] to [ g ∞ i , π ] and fix [ a ∞ , a ∞ , g ∞ i , → [ a ∞ ,
0] as i → ∞ ([7, p.187]).Here in this example, we note that(a) the point a i · x is in the geodesic segment from x to g i · x in X ,i.e., a i · x ∈ [ x , g i · x ] in X for any i ∈ N and(b) the distance between the point a i ∗ y and the geodesic segment from y to g i ∗ y is unbounded for i ∈ N in Y , i.e., there does not exist aconstant M > d ( a i ∗ y , [ y , g i ∗ y ]) ≤ M for any i ∈ N in Y .Based on this observation, we consider a condition.We suppose that a group G acts geometrically on two CAT(0) spaces X and Y . Let x ∈ X and y ∈ Y . Then we define the condition ( ∗ ) as follows:( ∗ ) There exist constants N >
M > GB ( x , N ) = X , GB ( y , M ) = Y and for any g, a ∈ G , if [ x , gx ] ∩ B ( ax , N ) = ∅ in X then [ y , gy ] ∩ B ( ay , M ) = ∅ in Y .We first prove the following theorem in Section 3. Theorem 1.1.
Suppose that a group G acts geometrically on two CAT(0)spaces X and Y . Let x ∈ X and y ∈ Y . If the condition ( ∗ ) holds, thenthere exists a G -equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuous extension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy . Next, we consider the following condition ( ∗∗ ).We suppose that a group G acts geometrically on two CAT(0) spaces X and Y . Let x ∈ X and y ∈ Y . Then we define the condition ( ∗∗ ) asfollows:( ∗∗ ) For any sequence { g i | i ∈ N } ⊂ G , the sequence { g i x | i ∈ N } is aCauchy sequence in X ∪ ∂X if and only if the sequence { g i y | i ∈ N } is a Cauchy sequence in Y ∪ ∂Y . hen we show the following theorem in Section 4. Theorem 1.2.
Suppose that a group G acts geometrically on two CAT(0)spaces X and Y . Let x ∈ X and y ∈ Y . The condition ( ∗∗ ) holds if andonly if there exists a G -equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuous extension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy . There are problems of rigidity in group actions (see Section 10).In this paper, a CAT(0) group G is said to be (boundary-)rigid , if G determines its ideal boundary up to homeomorphisms, i.e., all boundariesof CAT(0) spaces on which G acts geometrically are homeomorphic. Also aCAT(0) group G is said to be equivariant (boundary) rigid , if G determinesits ideal boundary by the equivariant homeomorphisms as above, i.e., if forany two CAT(0) spaces X and Y on which G acts geometrically it obtain a G -equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuousextension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy ,where x ∈ X and y ∈ Y .As an application of Theorem 1.1, we introduce some examples of equi-variant rigid CAT(0) groups in Section 5. In particular, any group of theform Z n ∗ · · · ∗ Z n k ∗ A ∗ · · · ∗ A l where n i ∈ N and each A j is a finite group is an equivariant rigid CAT(0)group. (Here we note that there is a recent research by G. C. Hruska [36]on CAT(0) spaces with isolated flats.)Also as an application of Theorem 1.2, we introduce an example of nonequivariant rigid CAT(0) groups in Section 6. In particular, we show thatevery CAT(0) group of the form G = F × H where F is a free group of rank n ≥ H is an infinite CAT(0) group, is non equivariant rigid.In Section 7, we provide some remarks and questions on (equivariant)rigidity of boundaries of CAT(0) groups. Here the following natural openproblem is important. Problem. If G and G are (equivariant) rigid CAT(0) groups, then is G ∗ G also?In Section 8, we investigate Coxeter groups acting on CAT(0) spaces asreflection groups.Now we consider a cocompact discrete reflection group G of a CAT(0)space X . (Here definitions and details of reflections and cocompact discretereflection groups of geodesic spaces are found in [32] and [34].) Then thegroup G becomes a Coxeter group.In this paper, a Coxeter group W is said to be equivariant rigid as areflection group , if for any two CAT(0) spaces X and Y on which W actsgeometrically as W becomes a cocompact discrete reflection group of X and Y , it obtain a W -equivariant homeomorphism of the boundaries ∂X and ∂Y s a continuous extension of the quasi-isometry φ : W x → W y defined by φ ( wx ) = wy , where x ∈ X and y ∈ Y .Then we show the following theorem. Theorem 1.3.
The following statements hold. (i)
If Coxeter groups W and W are equivariant rigid as reflectiongroups, then W ∗ W is also. (ii) For a Coxeter group W = W A ∗ W A ∩ B W B where W A ∩ B is finite, if W determines its Coxeter system up to isomorphism, and if W A and W B are equivariant rigid as reflection groups then W is also, where W T is the parabolic subgroup of W generated by T . Finally, by observing examples and applications of Theorems 1.1 and 1.2,the following conjecture arises.
Conjecture.
The group G = ( F × Z ) ∗ Z will be a non-rigid CAT(0) groupwith uncountably many boundaries.We explain where this conjecture comes from in Section 9 and introduceproblems of rigidity on group actions in Section 10.2. CAT(0) spaces and their boundaries
Details of CAT(0) spaces and their boundaries are found in [1], [8], [22],[23] and [50].A proper geodesic space (
X, d X ) is called a CAT(0) space , if the “CAT(0)-inequality” holds for all geodesic triangles △ and for all choices of two points x and y in △ . Here the “CAT(0)-inequality” is defined as follows: Let △ bea geodesic triangle in X . A comparison triangle for △ is a geodesic triangle △ ′ in the Euclidean plain R with same edge lengths as △ . Choose twopoints x and y in △ . Let x ′ and y ′ denote the corresponding points in △ ′ .Then the inequality d X ( x, y ) ≤ d R ( x ′ , y ′ )is called the CAT(0)-inequality , where d R is the natural metric on R .Every proper CAT(0) space can be compactified by adding its “bound-ary”. Let ( X, d X ) be a proper CAT(0) space, and let R be the set of allgeodesic rays in X . We define an equivalence relation ∼ in R as follows:For geodesic rays ξ, ζ : [0 , ∞ ) → X , ξ ∼ ζ ⇐⇒ Im ξ ⊂ B (Im ζ, N ) for some N ≥ , where B ( A, N ) := { x ∈ X | d X ( x, A ) ≤ N } for A ⊂ X . Then the boundary ∂X of X is defined as ∂X = R / ∼ . For each geodesic ray ξ ∈ R , the equivalence class of ξ is denoted by ξ ( ∞ ).It is known that for each α ∈ ∂X and each x ∈ X , there exists a uniquegeodesic ray ξ α : [0 , ∞ ) → X such that ξ α (0) = x and ξ α ( ∞ ) = α . Thus e can identify the boundary ∂X of X as the set of all geodesic rays ξ with ξ (0) = x .Let ( X, d X ) be a proper CAT(0) space and let x ∈ X . We define atopology on X ∪ ∂X as follows:(1) X is an open subspace of X ∪ ∂X .(2) Let α ∈ ∂X and let ξ α be the geodesic ray such that ξ α (0) = x and ξ α ( ∞ ) = α . For r > ǫ >
0, we define U X ∪ ∂X ( α ; r, ǫ ) = { x ∈ X ∪ ∂X | x B ( x , r ) , d X ( ξ α ( r ) , ξ x ( r )) < ǫ } , where ξ x : [0 , d X ( x , x )] → X is the geodesic (segment or ray) from x to x . Let ǫ > { U X ∪ ∂X ( α ; r, ǫ ) | r > } is a neighborhood basis for α in X ∪ ∂X .Here it is known that the topology on X ∪ ∂X is not dependent on thebasepoint x ∈ X and X ∪ ∂X is a metrizable compactification of X .Also for α ∈ ∂X and the geodesic ray ξ α with ξ α (0) = x and ξ α ( ∞ ) = α and for r > ǫ >
0, we define U ′ X ∪ ∂X ( α ; r, ǫ ) = { x ∈ X ∪ ∂X | x B ( x , r ) , d X ( ξ α ( r ) , Im ξ x ) < ǫ } , where ξ x : [0 , d ( x , x )] → X is the geodesic (segment or ray) from x to x .Let ǫ > { U ′ X ∪ ∂X ( α ; r, ǫ ) | r > } is also a neighborhood basis for α in X ∪ ∂X (cf. [31, Lemma 4.2]).Suppose that a group G acts on a proper CAT(0) space X by isometries.For each element g ∈ G and each geodesic ray ξ : [0 , ∞ ) → X , a map gξ : [0 , ∞ ) → X defined by ( gξ )( t ) := g ( ξ ( t )) is also a geodesic ray. Fortwo geodesic rays ξ and ξ ′ , if ξ ( ∞ ) = ξ ′ ( ∞ ) then gξ ( ∞ ) = gξ ′ ( ∞ ). Thus g induces a homeomorphism of ∂X , and G acts on ∂X by homeomorphisms.Here we note that if a sequence { x i | i ∈ N } ⊂ X converges to α ∈ ∂X in X ∪ ∂X , then for any g ∈ G , the sequence { gx i | i ∈ N } ⊂ X converges to gα ∈ ∂X in X ∪ ∂X . Definition 2.1.
Let (
X, d X ) be a proper CAT(0) space and let { x i | i ∈ N } ⊂ X be an unbounded sequence in X . In this paper, we say that thesequence { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X , if there exists ǫ > r >
0, there is a number i ∈ N as x i ∈ U X ∪ ∂X ( x i ; r, ǫ )for any i ≥ i . Here U X ∪ ∂X ( x i ; r, ǫ ) = { x ∈ X | x B ( x , r ) , d X ( ξ x i ( r ) , ξ x ( r )) < ǫ } , where ξ z is the geodesic segment from x to z in X .We show the following lemma used later. emma 2.2. Let ( X, d X ) be a proper CAT(0) space and let { x i | i ∈ N } ⊂ X be an unbounded sequence in X . Then the sequence { x i | i ∈ N } is a Cauchysequence in X ∪ ∂X defined above if and only if the sequence { x i | i ∈ N } converges to some point α ∈ ∂X in X ∪ ∂X .Proof. We first show that if the sequence { x i | i ∈ N } converges to somepoint α ∈ ∂X in X ∪ ∂X , then { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X defined above.Suppose that { x i | i ∈ N } converges to α ∈ ∂X in X ∪ ∂X . Let ǫ > { U X ∪ ∂X ( α ; r, ǫ | r > } is a neighborhood basis for α in X ∪ ∂X , for each r >
0, there exists anumber i ∈ N such that x i ∈ U X ∪ ∂X ( α ; r, ǫ i ≥ i . Then for any i ≥ i , d X ( ξ x i ( r ) , ξ x ( r )) ≤ d X ( ξ x i ( r ) , ξ α ( r )) + d X ( ξ α ( r ) , ξ x ( r )) ≤ ǫ ǫ ǫ . Hence x i ∈ U X ∪ ∂X ( x i ; r, ǫ ) for any i ≥ i . Thus the sequence { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X .Next, we show that if { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X definedabove, then { x i | i ∈ N } converges to some point α ∈ ∂X in X ∪ ∂X .Suppose that { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X . Since the set { x i | i ∈ N } is unbounded in X , there exists a limit point α ∈ Cl { x i | i ∈ N } ∩ ∂X . Here there exists a subsequence { x i j | j ∈ N } ⊂ { x i | i ∈ N } whichconverges to α in X ∪ ∂X .Then we show that the sequence { x i | i ∈ N } converges to the point α ∈ ∂X in X ∪ ∂X .Since { x i | i ∈ N } is a Cauchy sequence in X ∪ ∂X , there exists ǫ > r >
0, there is a number i ∈ N as x i ∈ U X ∪ ∂X ( x i ; r, ǫ )for any i ≥ i , i.e., d X ( ξ x i ( r ) , ξ x i ( r )) ≤ ǫ for any i ≥ i . Also since thesubsequence { x i j | j ∈ N } converges to α in X ∪ ∂X , there exists i j ≥ i such that x i j ∈ U X ∪ ∂X ( α ; r, d X ( ξ x ij ( r ) , ξ α ( r )) ≤
1. Then for any i ≥ i j , d X ( ξ x i ( r ) , ξ α ( r )) ≤ d X ( ξ x i ( r ) , ξ x i ( r )) + d X ( ξ x i ( r ) , ξ x ij ( r )) + d X ( ξ x ij ( r ) , ξ α ( r )) ≤ ǫ + ǫ + 1= 2 ǫ + 1 , since i ≥ i j ≥ i . Hence for any r >
0, there exists a number i j ∈ N suchthat for any i ≥ i j , x i ∈ U X ∪ ∂X ( α ; r, ǫ + 1) , here 2 ǫ + 1 is a constant. Thus the sequence { x i | i ∈ N } converges to thepoint α ∈ ∂X in X ∪ ∂X . (cid:3) Proof of Theorem 1.1
In this section, we prove Theorem 1.1.We suppose that a group G acts geometrically on two CAT(0) spaces( X, d X ) and ( Y, d Y ). Let x ∈ X and y ∈ Y . Now we suppose that thecondition ( ∗ ) holds; that is,( ∗ ) there exist constants N >
M > GB ( x , N ) = X , GB ( y , M ) = Y and for any g, a ∈ G , if [ x , gx ] ∩ B ( ax , N ) = ∅ in X then [ y , gy ] ∩ B ( ay , M ) = ∅ in Y .Our goal in this section is to show that the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy continuously extends to a G -equivariant homeo-morphism of the boundaries ∂X and ∂Y .Since the map φ : Gx → Gy defined by φ ( gx ) = gy is a quasi-isometry(cf. [8, p.138], [24], [25]), there exist constants λ > C > λ d Y ( gy , hy ) − C ≤ d X ( gx , hx ) ≤ λd Y ( gy , hy ) + C for any g, h ∈ G .We first show the following. Proposition 3.1.
Let { g i } ⊂ G be a sequence. If { g i x } ⊂ X is a Cauchysequence in X ∪ ∂X defined in Section 2, then { g i y } ⊂ Y is also a Cauchysequence in Y ∪ ∂Y .Proof. Let { g i } ⊂ G . Suppose that { g i x } ⊂ X is a Cauchy sequence in X ∪ ∂X .To prove that { g i y } ⊂ Y is a Cauchy sequence in Y ∪ ∂Y , we show thatthere exists M ′ > R >
0, there is i ∈ N as g i y ∈ U Y ∪ ∂Y ( g i y ; R, M ′ )for any i ≥ i .Let M ′ = λ (2 N + 1) + 2 M + C and let R > { g i x } ⊂ X is a Cauchy sequence in X ∪ ∂X , for r = λ ( R + C + M ) + N , there exists i ∈ N such that g i x ∈ U X ∪ ∂X ( g i x ; r, i ≥ i .Then d X ( x , g i x ) ≥ r, d X ( x , g i x ) ≥ r, and d X ( ξ g i x ( r ) , ξ g i x ( r )) ≤ , where ξ g i x is the geodesic from x to g i x and ξ g i x is the geodesic from x to g i x in X . ince GB ( x , N ) = X , there exist a, b ∈ G such that d X ( ax , ξ g i x ( r )) ≤ N and d X ( bx , ξ g i x ( r )) ≤ N . Then[ x , g i x ] ∩ B ( ax , N ) = ∅ and [ x , g i x ] ∩ B ( bx , N ) = ∅ . Hence by the condition ( ∗ ),[ y , g i y ] ∩ B ( ay , M ) = ∅ and [ y , g i y ] ∩ B ( by , M ) = ∅ . Thus ξ g i y ( r ′ ) ∈ [ y , g i y ] ∩ B ( ay , M ) and ξ g i y ( r ′ ) ∈ [ y , g i y ] ∩ B ( by , M )for some r ′ > r ′ > i ≥ i , g i y ∈ U Y ∪ ∂Y ( g i y ; R, M ′ ) , we show that r ′ ≥ R, r ′ ≥ R and d Y ( ξ g i y ( r ′ ) , ξ g i y ( r ′ )) ≤ M ′ . First, r ′ = d Y ( y , ξ g i y ( r ′ )) ≥ d Y ( y , ay ) − M ≥ λ d X ( x , ax ) − C − M ≥ λ ( r − N ) − C − M = R, because d X ( x , ax ) ≥ r − N and r = λ ( R + C + M ) + N .Similary, r ′ ≥ R , because d X ( x , bx ) ≥ r − N and r = λ ( R + C + M )+ N .Also, d Y ( ξ g i y ( r ′ ) , ξ g i y ( r ′ )) ≤ d Y ( ay , by ) + 2 M ≤ ( λd X ( ax , bx ) + C ) + 2 M ≤ λ ( d X ( ξ g i x ( r ) , ξ g i x ( r )) + 2 N ) + C + 2 M ≤ λ (1 + 2 N ) + C + 2 M = M ′ , because d X ( ξ g i x ( r ) , ξ g i x ( r )) ≤ M ′ = λ (2 N + 1) + 2 M + C .Thus r ′ ≥ R, r ′ ≥ R and d Y ( ξ g i y ( r ′ ) , ξ g i y ( r ′ )) ≤ M ′ . Hence d Y ( ξ g i y ( R ) , ξ g i y ( R )) ≤ d Y ( ξ g i y ( r ′ ) , ξ g i y ( r ′ )) ≤ M ′ , since Y is a CAT(0) space. Also we obtain that d Y ( y , g i y ) ≥ R and d Y ( y , g i y ) ≥ R, because r ′ ≥ R and r ′ ≥ R . hus g i y ∈ U Y ∪ ∂Y ( g i y ; R, M ′ )for any i ≥ i . Hence we obtain that { g i y } ⊂ Y is a Cauchy sequence in Y ∪ ∂Y . (cid:3) Then we can define a map ¯ φ : ∂X → ∂Y as a continuous extension ofthe quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy as follows: Foreach α ∈ ∂X , there exists a sequence { g i x } ⊂ Gx ⊂ X which convergesto α in X ∪ ∂X . Then the sequence { g i x } ⊂ X is a Cauchy sequence in X ∪ ∂X by Lemma 2.2. By Proposition 3.1, the sequence { g i y } ⊂ Y isalso a Cauchy sequence in Y ∪ ∂Y . Hence by Lemma 2.2, the sequence { g i y } ⊂ Y converges to some point ¯ α ∈ ∂Y in Y ∪ ∂Y . Then we define¯ φ ( α ) = ¯ α . Proposition 3.2.
The map ¯ φ : ∂X → ∂Y is well-defined.Proof. Let α ∈ ∂X and let { g i x } , { h i x } ⊂ Gx ⊂ X be two sequenceswhich converge to α in X ∪ ∂X . As the argument above, by Lemma 2.2 andProposition 3.1, the sequence { g i y } ⊂ Y converges to some point ¯ α ∈ ∂Y and the sequence { h i y } ⊂ Y converges to some point ¯ β ∈ ∂Y in Y ∪ ∂Y .Then we show that ¯ α = ¯ β .Here we can consider a sequence { ˜ g j x | j ∈ N } ⊂ Gx ⊂ X such that { ˜ g j x | j ∈ N } = { g i x | i ∈ N } ∪ { h i x | i ∈ N } and the sequence { ˜ g j x } converges to α in X ∪ ∂X . Then the sequence { ˜ g j x } is a Cauchy sequence in X ∪ ∂X and the sequence { ˜ g j y } is also in Y ∪ ∂Y by Proposition 3.1. Hence the sequence { ˜ g j y } converges to somepoint ¯ γ ∈ ∂Y in Y ∪ ∂Y . Here we note that the two sequences { g i y } and { h i y } are subsequences of { ˜ g j y } . Hence we obtain that ¯ α = ¯ β = ¯ γ .Thus the map ¯ φ : ∂X → ∂Y defined as above is well-defined. (cid:3) Next, we show the following.
Proposition 3.3.
The map ¯ φ : ∂X → ∂Y is surjective.Proof. Let ¯ α ∈ ∂Y . There exists a sequence { g i y } ⊂ Gy ⊂ Y whichconverges to ¯ α in Y ∪ ∂Y . Then we consider the set { g i x | i ∈ N } which isan unbounded subset of X . HereCl { g i x | i ∈ N } ∩ ∂X = ∅ , and there exists a subsequence { g i j x | j ∈ N } ⊂ { g i x } which converges tosome point α ∈ ∂X . Then the sequence { g i j y } converges to ¯ α in Y ∪ ∂Y ,because { g i j y } is a subsequence of the sequence { g i y } which converges to¯ α in Y ∪ ∂Y . Hence ¯ φ ( α ) = ¯ α by the definition of the map ¯ φ . Thus the map¯ φ : ∂X → ∂Y is surjective. (cid:3) Here we provide a lemma. emma 3.4. For any ˜ N ≥ N , there exists ˜ M > such that GB ( y , ˜ M ) = Y and for any g, a ∈ G , if [ x , gx ] ∩ B ( ax , ˜ N ) = ∅ in X then [ y , gy ] ∩ B ( ay , ˜ M ) = ∅ in Y .Proof. For ˜ N ≥ N , we put ˜ M = λ ( N + ˜ N ) + C + M .Let g, a ∈ G as [ x , gx ] ∩ B ( ax , ˜ N ) = ∅ in X . Then there exists apoint x ∈ [ x , gx ] ∩ B ( ax , ˜ N ). Since GB ( x , N ) = X , there exists a ′ ∈ G such that x ∈ B ( a ′ x , N ). Then x ∈ [ x , gx ] ∩ B ( a ′ x , N ) and [ x , gx ] ∩ B ( a ′ x , N ) = ∅ in X . By the condition ( ∗ ), [ y , gy ] ∩ B ( a ′ y , M ) = ∅ in Y .Hence d Y ( a ′ y , [ y , gy ]) ≤ M . Here we note that d Y ( a ′ y , ay ) ≤ λd X ( a ′ x , ax ) + C ≤ λ ( d X ( a ′ x , x ) + d X ( x , ax )) + C ≤ λ ( N + ˜ N ) + C. Hence d Y ( ay , [ y , gy ]) ≤ d Y ( ay , a ′ y ) + d Y ( a ′ y , [ y , gy ]) ≤ λ ( N + ˜ N ) + C + M = ˜ M .
Thus we obtain that [ y , gy ] ∩ B ( ay , ˜ M ) = ∅ in Y . (cid:3) Let ˜ N = 2 N . By Lemma 3.4, there exists ˜ M > GB ( y , ˜ M ) = Y and for any g, a ∈ G , if [ x , gx ] ∩ B ( ax , ˜ N ) = ∅ in X then [ y , gy ] ∩ B ( ay , ˜ M ) = ∅ in Y .Here we show the following technical lemma. Lemma 3.5.
Let α ∈ ∂X and let ξ α : [0 , ∞ ) → X be the geodesic ray in X such that ξ α (0) = x and ξ α ( ∞ ) = α . Let { g i x } ⊂ Gx ⊂ X be a sequencewhich converges to α in X ∪ ∂X such that d X ( g i x , ξ α ( i )) ≤ N for any i ∈ N (since GB ( x , N ) = X , we can take such a sequence). Then (1) d X ( g i x , [ x , g j x ]) ≤ ˜ N for any i, j ∈ N with i < j , (2) d Y ( g i y , [ y , g j y ]) ≤ ˜ M for any i, j ∈ N with i < j , (3) d Y ( g i y , Im ξ ¯ α ) ≤ ˜ M + 1 for any i ∈ N , (4) d X ( g i x , g i +1 x ) ≤ N + 1 for any i ∈ N , (5) d Y ( g i y , g i +1 y ) ≤ λ (2 N + 1) + C for any i ∈ N , and (6) Im ξ ¯ α ⊂ S { B ( g i y ,
3( ˜ M + 1) + λ (2 N + 1) + C ) | i ∈ N } .Here ¯ α = ¯ φ ( α ) and ξ ¯ α : [0 , ∞ ) → Y is the geodesic ray in Y such that ξ ¯ α (0) = y and ξ ¯ α ( ∞ ) = ¯ α .Proof. (1) For any i, j ∈ N with i < j , d X ( g i x , [ x , g j x ]) ≤ d X ( g i x , ξ α ( i )) + d X ( ξ α ( i ) , [ x , g j x ]) ≤ N + N = 2 N = ˜ N , here we obtain the inequality d X ( ξ α ( i ) , [ x , g j x ]) ≤ N , since d X ( g j x , ξ α ( j )) ≤ N , i < j and X is a CAT(0) space.(2) By Lemma 3.4 and the definition of ˜ M , we obtain that d Y ( g i y , [ y , g j y ]) ≤ ˜ M for any i, j ∈ N with i < j from (1).(3) We note that the sequence { g i y } converges to ¯ α by the definition ofthe map ¯ φ : ∂X → ∂Y .Let i ∈ N and let R = d Y ( y , g i y ). Since the sequence { g j y } convergesto ¯ α , there exists j ∈ N such that d Y ( ξ ¯ α ( R ) , ξ g j y ( R )) < j ≥ j , because the set { U Y ∪ ∂Y ( ¯ α ; r, | r > } defined in Section 2 is a neighborhood basis for ¯ α in Y ∪ ∂Y .Let j ∈ N with j > i and j > j . Since i < j , we obtain that d Y ( g i y , [ y , g j y ]) ≤ ˜ M by (2). Hence there exists r > d Y ( g i y , ξ g j y ( r )) ≤ ˜ M . Here we note that r ≤ R by [31, Lemma 4.1]and we can obtain that d Y ( ξ ¯ α ( r ) , ξ g j y ( r )) < d Y ( ξ ¯ α ( R ) , ξ g j y ( R )) < , since Y is a CAT(0) space. Then d Y ( g i y , Im ξ ¯ α ) ≤ d Y ( g i y , ξ g j y ( r )) + d Y ( ξ g j y ( r ) , Im ξ ¯ α ) < ˜ M + 1 . Hence d Y ( g i y , Im ξ ¯ α ) ≤ ˜ M + 1 for any i ∈ N .(4) We obtain that d X ( g i x , g i +1 x ) ≤ N + 1 for any i ∈ N , because d X ( g i x , g i +1 x ) ≤ d X ( g i x , ξ α ( i )) + d X ( ξ α ( i ) , ξ α ( i + 1)) + d X ( ξ α ( i + 1) , g i +1 x ) ≤ N + 1 + N = 2 N + 1 , since d X ( g i x , ξ α ( i )) ≤ N for any i ∈ N by the definition of the sequence { g i x } .(5) Since the map φ : Gx → Gy ( gx gy ) is a quasi-isometry, weobtain that d Y ( g i y , g i +1 y ) ≤ λ (2 N + 1) + C for any i ∈ N by (4).(6) For each i ∈ N , there exists r i > d Y ( g i y , ξ ¯ α ( r i )) ≤ ˜ M + 1by (3). Then by (5), d Y ( ξ ¯ α ( r i ) , ξ ¯ α ( r i +1 )) ≤ d Y ( ξ ¯ α ( r i ) , g i y ) + d Y ( g i y , g i +1 y ) + d Y ( g i +1 y , ξ ¯ α ( r i +1 )) ≤ ( ˜ M + 1) + ( λ (2 N + 1) + C ) + ( ˜ M + 1)= 2( ˜ M + 1) + λ (2 N + 1) + C. Hence we obtain thatIm ξ ¯ α ⊂ [ { B ( g i y ,
3( ˜ M + 1) + λ (2 N + 1) + C ) | i ∈ N } . (cid:3) ow we show the following. Proposition 3.6.
The map ¯ φ : ∂X → ∂Y is injective.Proof. Let α, α ′ ∈ ∂X , and let ξ α : [0 , ∞ ) → X and ξ α ′ : [0 , ∞ ) → X be the geodesic rays in X such that ξ α (0) = ξ α ′ (0) = x , ξ α ( ∞ ) = α and ξ α ′ ( ∞ ) = α ′ . Let { g i x } , { g ′ i x } ⊂ Gx ⊂ X be sequences such that d X ( g i x , ξ α ( i )) ≤ N and d X ( g ′ i x , ξ α ′ ( i )) ≤ N . Then the sequence { g i x } converges to α and the sequence { g ′ i x } converges to α ′ in X ∪ ∂X .Let ¯ α = ¯ φ ( α ) and ¯ α ′ = ¯ φ ( α ′ ). Also let ξ ¯ α : [0 , ∞ ) → Y and ξ ¯ α ′ : [0 , ∞ ) → Y be the geodesic rays in Y such that ξ ¯ α (0) = ξ ¯ α ′ (0) = y , ξ ¯ α ( ∞ ) = ¯ α and ξ ¯ α ′ ( ∞ ) = ¯ α ′ .Then by Lemma 3.5,(1) d X ( g i x , [ x , g j x ]) ≤ ˜ N for any i, j ∈ N with i < j ,(2) d Y ( g i y , [ y , g j y ]) ≤ ˜ M for any i, j ∈ N with i < j ,(3) d Y ( g i y , Im ξ ¯ α ) ≤ ˜ M + 1 for any i ∈ N ,(4) d X ( g i x , g i +1 x ) ≤ N + 1 for any i ∈ N ,(5) d Y ( g i y , g i +1 y ) ≤ λ (2 N + 1) + C for any i ∈ N ,(6) Im ξ ¯ α ⊂ S { B ( g i y ,
3( ˜ M + 1) + λ (2 N + 1) + C ) | i ∈ N } ,and(1 ′ ) d X ( g ′ i x , [ x , g ′ j x ]) ≤ ˜ N for any i, j ∈ N with i < j ,(2 ′ ) d Y ( g ′ i y , [ y , g ′ j y ]) ≤ ˜ M for any i, j ∈ N with i < j ,(3 ′ ) d Y ( g ′ i y , Im ξ ¯ α ′ ) ≤ ˜ M + 1 for any i ∈ N ,(4 ′ ) d X ( g ′ i x , g ′ i +1 x ) ≤ N + 1 for any i ∈ N ,(5 ′ ) d Y ( g ′ i y , g ′ i +1 y ) ≤ λ (2 N + 1) + C for any i ∈ N ,(6 ′ ) Im ξ ¯ α ′ ⊂ S { B ( g ′ i y ,
3( ˜ M + 1) + λ (2 N + 1) + C ) | i ∈ N } .To prove that the map ¯ φ : ∂X → ∂Y is injective, we show that if α = α ′ then ¯ α = ¯ α ′ .We suppose that α = α ′ . Then the geodesic rays ξ α and ξ α ′ arenot asymptotic. Hence for any t >
0, there exists r > d X ( ξ α ( r ) , Im ξ α ′ ) > t . Then for i ∈ N with i ≥ r , d X ( g i x , Im ξ α ′ ) ≥ d X ( ξ α ( i ) , Im ξ α ′ ) − d X ( g i x , ξ α ( i )) ≥ d X ( ξ α ( r ) , Im ξ α ′ ) − N> t − N Since d X ( g ′ j x , Im ξ α ′ ) ≤ N for any j ∈ N , we obtain that d X ( g i x , g ′ j x ) >t − N for any j ∈ N . Hence for any j ∈ N , d Y ( g i y , g ′ j y ) ≥ λ d X ( g i x , g ′ j x ) − C> λ ( t − N ) − C. Here by (6 ′ ),Im ξ ¯ α ′ ⊂ [ { B ( g ′ j y ,
3( ˜ M + 1) + λ (2 N + 1) + C ) | j ∈ N } . et j ∈ N such that d Y ( g i y , g ′ j y ) = min { d Y ( g i y , g ′ j y ) | j ∈ N } . Then d Y ( g i y , Im ξ ¯ α ′ ) ≥ min { d Y ( g i y , g ′ j y ) | j ∈ N } − (3( ˜ M + 1) + λ (2 N + 1) + C )= d Y ( g i y , g ′ j y ) − (3( ˜ M + 1) + λ (2 N + 1) + C ) > ( 1 λ ( t − N ) − C ) − (3( ˜ M + 1) + λ (2 N + 1) + C ) , since d Y ( g i y , g ′ j y ) > λ ( t − N ) − C for any j ∈ N by the argument above.Thus for any t >
0, there exists i ∈ N such that d Y ( g i y , Im ξ ¯ α ′ ) > ( 1 λ ( t − N ) − C ) − (3( ˜ M + 1) + λ (2 N + 1) + C ) . Here by (3), there exists R > d Y ( g i y , ξ ¯ α ( R )) ≤ ˜ M + 1 . Then d Y ( ξ ¯ α ( R ) , Im ξ ¯ α ′ ) ≥ d Y ( g i y , Im ξ ¯ α ′ ) − d Y ( g i y , ξ ¯ α ( R )) > ( 1 λ ( t − N ) − C ) − (3( ˜ M + 1) + λ (2 N + 1) + C ) − ( ˜ M + 1)= ( 1 λ ( t − N ) − C ) − (4( ˜ M + 1) + λ (2 N + 1) + C ) . Since t > ξ ¯ α and ξ ¯ α ′ are not asymptotic and ¯ α = ¯ α ′ .Therefore, the map ¯ φ : ∂X → ∂Y is injective. (cid:3) From Propositions 3.3 and 3.6, we obtain that the map ¯ φ : ∂X → ∂Y isbijective.Then we show the following. Proposition 3.7.
The map ¯ φ : ∂X → ∂Y is continuous.Proof. Let α ∈ ∂X and let { α i | i ∈ N } ⊂ ∂X be a sequence which convergesto α in ∂X .It is sufficient to show that the sequence { ¯ α i | i ∈ N } converges to ¯ α in ∂Y , where ¯ α i = ¯ φ ( α i ) and ¯ α = ¯ φ ( α ).For each i ∈ N , there exists a sequence { a i,j x | j ∈ N } which convergesto α i as j → ∞ .Since the sequence { α i | i ∈ N } ⊂ ∂X converges to α in ∂X , for any r > i ∈ N such that α i ∈ U X ∪ ∂X ( α ; r, i ≥ i . Since the sequence { a i,j x | j ∈ N } converges to α i as j → ∞ in X ∪ ∂X , for any i ≥ i , there exists j i ∈ N such that a i,j x ∈ U X ∪ ∂X ( α i ; r, or any j ≥ j i .Then a i,j x ∈ U X ∪ ∂X ( α ; r, i ≥ i and j ≥ j i .Now the sequence { a i,j x | j ≥ j i , i ≥ i } converges to α as i → ∞ in X ∪ ∂X . Hence the sequence { a i,j y | j ≥ j i , i ≥ i } converges to ¯ φ ( α ) = ¯ α as i → ∞ in Y ∪ ∂Y .Here we note that the sequence { a i,j y | j ∈ N } converges to ¯ φ ( α i ) = ¯ α i as j → ∞ in Y ∪ ∂Y .Then for any r >
0, there exists an enough large number i ≥ i such that a i,j y ∈ U Y ∪ ∂Y ( ¯ α ; r, j ≥ j i . Also there exists an enough large number j ≥ j i such that a i,j y ∈ U Y ∪ ∂Y ( ¯ α i ; r, . Then we obtain that ¯ α i ∈ U Y ∪ ∂Y ( ¯ α ; r, . Thus the sequence { ¯ α i | i ∈ N } converges to ¯ α in ∂Y and the map ¯ φ : ∂X → ∂Y is continuous. (cid:3) Therefore we obtain the following.
Theorem 3.8.
The map ¯ φ : ∂X → ∂Y is a G -equivariant homeomorphism.Proof. By the argument above, the map ¯ φ : ∂X → ∂Y is well-defined,bijective and continuous.From the definition and the well-definedness of ¯ φ , we obtain that the map¯ φ : ∂X → ∂Y is G -equivariant. Indeed for any α ∈ ∂X and g ∈ G , if { g i x } ⊂ Gx ⊂ X is a sequence which converges to α in X ∪ ∂X , then¯ φ ( α ) is the point of ∂Y to which the sequence { g i y } ⊂ Gy ⊂ Y convergesin Y ∪ ∂Y . Then { gg i x } ⊂ Gx ⊂ X is the sequence which convergesto gα in X ∪ ∂X and ¯ φ ( gα ) is the point of ∂Y to which the sequence { gg i y } ⊂ Gy ⊂ Y converges in Y ∪ ∂Y . Here we note that the sequence { gg i y } ⊂ Gy ⊂ Y converges to g ¯ φ ( α ) in Y ∪ ∂Y by the definition of theaction of G on ∂Y . Hence ¯ φ ( gα ) = g ¯ φ ( α ) for any α ∈ ∂X and g ∈ G andthe map ¯ φ : ∂X → ∂Y is G -equivariant.Also, the map ¯ φ : ∂X → ∂Y is closed, since ∂X and ∂Y are compact andmetrizable.Therefore, we obtain that the map ¯ φ : ∂X → ∂Y is a G -equivarianthomeomorphism. (cid:3) Proof of Theorem 1.2
In this section, we prove Theorem 1.2.We suppose that a group G acts geometrically on two CAT(0) spaces( X, d X ) and ( Y, d Y ). Let x ∈ X and y ∈ Y . e first show that if the condition ( ∗∗ ) holds, then there exists a G -equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuousextension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy .Now we suppose that the condition ( ∗∗ ) holds; that is,( ∗∗ ) For any sequence { g i | i ∈ N } ⊂ G , the sequence { g i x | i ∈ N } is aCauchy sequence in X ∪ ∂X if and only if the sequence { g i y | i ∈ N } is a Cauchy sequence in Y ∪ ∂Y .We define a map ¯ φ : ∂X → ∂Y as a continuous extension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy by the same argumentin Section 3: For each α ∈ ∂X , there exists a sequence { g i x } ⊂ Gx ⊂ X which converges to α in X ∪ ∂X . Then the sequence { g i x } ⊂ X is a Cauchysequence in X ∪ ∂X . By the condition ( ∗∗ ), the sequence { g i y } ⊂ Y is alsoa Cauchy sequence in Y ∪ ∂Y . Hence the sequence { g i y } ⊂ Y converges tosome point ¯ α ∈ ∂Y in Y ∪ ∂Y . Then we define ¯ φ ( α ) = ¯ α .We obtain the following by the same proof as the one of Proposition 3.2. Proposition 4.1.
The map ¯ φ : ∂X → ∂Y is well-defined. Also we can define the inverse ¯ φ − : ∂Y → ∂X as a continuous extensionof the quasi-isometry φ − : Gy → Gx defined by φ − ( gy ) = gx by thecondition ( ∗∗ ).Hence, we obtain the following. Proposition 4.2.
The map ¯ φ : ∂X → ∂Y is bijective. Also, we obtain the following by the same proof as the one of Proposi-tion 3.7.
Proposition 4.3.
The map ¯ φ : ∂X → ∂Y is continuous. Thus we obtain the following theorem.
Theorem 4.4.
The map ¯ φ : ∂X → ∂Y is a G -equivariant homeomorphism. Conversely, we suppose that the condition ( ∗∗ ) does not hold.Then there exists a sequence { g i | i ∈ N } ⊂ G such that either(1) { g i x | i ∈ N } is a Cauchy sequence in X ∪ ∂X and { g i y | i ∈ N } is not a Cauchy sequence in Y ∪ ∂Y , or(2) { g i x | i ∈ N } is not a Cauchy sequence in X ∪ ∂X and { g i y | i ∈ N } is a Cauchy sequence in Y ∪ ∂Y .Now we suppose that (1) { g i x | i ∈ N } is a Cauchy sequence in X ∪ ∂X and { g i y | i ∈ N } is not a Cauchy sequence in Y ∪ ∂Y .Then, the sequence { g i x | i ∈ N } is Cauchy and converges to some point α ∈ ∂X in X ∪ ∂X . On the other hand, the sequence { g i y | i ∈ N } is notCauchy and contains subsequences { g i j y | j ∈ N } and { g k j y | j ∈ N } whichconverge to some points ¯ α and ¯ β in ∂Y respectively, with ¯ α = ¯ β .This means that the map φ : Gx → Gy ( gx gy ) does not continu-ously extend to any map ¯ φ : ∂X → ∂Y . y the same argument, we obtain that if (2) { g i x | i ∈ N } is not a Cauchysequence in X ∪ ∂X and { g i y | i ∈ N } is a Cauchy sequence in Y ∪ ∂Y , thenthe map φ − : Gy → Gx ( gy gx ) does not continuously extend to anymap ¯ φ − : ∂Y → ∂X .Therefore, there does not exist a G -equivariant homeomorphism of theboundaries ∂X and ∂Y as a continuous extension of the quasi-isometry φ : Gx → Gy defined by φ ( gx ) = gy .5. Examples of equivariant rigid CAT(0) groups
As an application of Theorem 1.1 and the condition ( ∗ ), we introducesome examples. Example 5.1.
Let G = Z × Z . The group G acts geometrically on the flatplane X = R by ( a, b ) · ( x, y ) = ( x + a, y + b ) for any ( a, b ) ∈ Z × Z = G and ( x, y ) ∈ R = X .Suppose that the group G acts geometrically on a CAT(0) space Y . Thenby the Flat Torus Theorem [8, Theorem II.7.1], there exist a quasi-convexsubset Y ′ of Y and a point y ∈ Y such that Y ′ is isometric to R and Y ′ is the convex hull C ( Gy ) of the orbit Gy in Y , where the orbit Gy is alattice in Y ′ ∼ = R .Then there exists a linear transformation φ : X → Y ′ .In this case, we see that the condition ( ∗ ) holds for the actions of thegroup G on X and Y .In fact, the induced map ¯ φ : ∂X → ∂Y of the boundaries ∂X and ∂Y that are homeomorphic to a circle S is an equivariant homeomorphism. Example 5.2.
By the same argument as above, we obtain that any geo-metric actions of a group G = Z n ( n ∈ N ) on any CAT(0) spaces satisfy thecondition ( ∗ ). Hence G = Z n is an equivariant rigid CAT(0) group. Example 5.3.
Let G = ( Z × Z ) ∗ Z which is the free product of Z × Z and Z . Let G = Z × Z and A = Z ; that is, G = G ∗ A .We construct a CAT(0) cubical cell complex Σ as follows: Let X = R on which G = Z × Z acts geometrically by ( a, b ) · ( x, y ) = ( x + a, y + b )for each ( a, b ) ∈ G and ( x, y ) ∈ X . Here we consider that X = R is thecubical cell complex whose 1-skeleton is the Cayley graph of G = Z × Z .Also let X = [0 ,
1] on which G = Z acts by ¯0 · x = x and ¯1 · x = 1 − x for each ¯0 , ¯1 ∈ Z and x ∈ [0 ,
1] = X ; that is, A = Z is a reflectiongroup of X = [0 , X = [0 ,
1] is the cubicalcomplex whose 1-skeleton is the Cayley graph of A = Z . Then we definethe 2-dimensional cubical cell complex Σ asΣ = [ { gX | g ∈ G } ∪ [ { gX | g ∈ G } , where we identify the two points g · (0 , ∈ gX and g · ∈ gX for any g ∈ G and the 1-skeleton of Σ is the Cayley graph of G = ( Z × Z ) ∗ Z . his construction is similar to one of the Davis complex of the right-angledCoxeter group W = (( Z ∗ Z ) × ( Z ∗ Z )) ∗ Z .Now we show that if the group G = ( Z × Z ) ∗ Z acts geometrically on aCAT(0) space Y , then the actions of G on Σ and Y satisfy the condition ( ∗ ).Suppose that the group G = G ∗ A = ( Z × Z ) ∗ Z acts geometricallyon a CAT(0) space Y . Then there exists y ∈ Y such that the convex-hull C ( G y ) is isometric to R by [8, Theorem II.7.1]. We put Y = C ( G y ).Let identify A = { , a } ( a = 1). Then we note that ∂Y ∩ ∂aY = ∅ .Hence there exists M > y, y ′ ] ∩ B ( y , M ) = ∅ and[ y, y ′ ] ∩ B ( ay , M ) = ∅ for any y ∈ Y and y ′ ∈ aY .Here for any g ∈ G , we can write g = g a · · · g n a n for some g i ∈ G and a i ∈ A (where it may g = 1 or a n = 1). Then[ x , gx ] = [ x , g x ] ∪ [ g x , g a x ] ∪ · · · ∪ [ g a · · · g n x , gx ] , in Σ. Also, [ y , gy ] ∩ B ( g a · · · g i − a i − g i y , M ) = ∅ and[ y , gy ] ∩ B ( g a · · · g i a i y , M ) = ∅ for any i = 1 , . . . , n in Y .Thus, we obtain that the geometric action of G on Σ and any geometricaction of G on any CAT(0) space Y satisfy the condition ( ∗ ).Therefore, the group G = ( Z × Z ) ∗ Z is an equivariant rigid CAT(0)group. Example 5.4.
By the same argument as above, we obtain that groups ofthe form G = Z n ∗ Z ( n ∈ N ) are equivariant rigid CAT(0) groups. Example 5.5.
Let G = Z n ∗ A which is the free product of Z n and a finitegroup A and let G = Z n ; that is, G = G ∗ A .We construct a CAT(0) cubical cell complex Σ as follows:Let X = R n on which G = Z n acts geometrically by ( a , . . . , a n ) · ( x , . . . , x n ) = ( x + a , . . . , x n + a n ) for any ( a , . . . , a n ) ∈ G and( x , . . . , x n ) ∈ X . Here we consider that X = R n is the cubical cell com-plex whose 1-skeleton is the Cayley graph of G = Z n . Then we considerthe set { gX | g ∈ G } , where each gX is a copy of X and gX = hX ifand only if g − h ∈ G . Let x := 0 in X = R n . Then G x is a lattice of X = R n .Also let X be the cone x ∗ Az of Az = { az | a ∈ A } where the lengthof [ x, az ] is 1 and [ x, az ] is isometric to [0 , A acts naturally byisometries on X by a · x = x and a · bz = abz for any a ∈ A and bz ∈ Az ⊂ X . We may consider that X is a 1-dimensional cubical complex.Then we consider the set { gX | g ∈ G } , where each gX is a copy of X and gX = hX if and only if g − h ∈ A . ere for each g ∈ G , we glue gX and gX by the one-point union gX ∨ gx = gz gX . Also we define the n -dimensional cubical cell complex Σ asΣ = [ { gX | g ∈ G } ∪ [ { gX | g ∈ G } , where we identify gx = gz for any g ∈ G . Then Σ is contractible, since G = G ∗ A is the free-product of G and A . Moreover, Σ is a CAT(0) spaceon which G = G ∗ A naturally acts geometrically.Now we show that if the group G = G ∗ A acts geometrically on a CAT(0)space Y , then the actions of G on Σ and Y satisfy the condition ( ∗ ).Suppose that the group G = G ∗ A acts geometrically on a CAT(0) space Y . Then there exists y ∈ Y such that the convex-hull C ( G y ) is isometricto R n by [8, Theorem II.7.1].Let a ∈ A − { } . Then we note that ∂Y ∩ ∂aY = ∅ . Hence, by thestructure of G , there exists M > y, y ′ ] ∩ B ( y , M ) = ∅ and[ y, y ′ ] ∩ B ( ay , M ) = ∅ for any y ∈ Y and y ′ ∈ aY .For any g ∈ G , we can write g = g a · · · g n a n for some g i ∈ G and a i ∈ A . Then[ x , gx ] = [ x , g x ] ∪ [ g x , g a x ] ∪ · · · ∪ [ g a · · · g n x , gx ] , in Σ. Also, [ y , gy ] ∩ B ( g a · · · g i − a i − g i y , M ) = ∅ and[ y , gy ] ∩ B ( g a · · · g i a i y , M ) = ∅ for any i = 1 , . . . , n in Y .Hence, we obtain that the geometric action of G on Σ and any geometricaction of G on any CAT(0) space Y satisfy the condition ( ∗ ).Therefore, the group G = Z n ∗ A is an equivariant rigid CAT(0) group. Example 5.6.
Groups of the form G = Z n ∗ Z n ( n , n ∈ N ) are equivari-ant rigid CAT(0) groups.First, for G = Z n ∗ Z n ( n , n ∈ N ) where we put G = Z n and G = Z n ; that is G = G ∗ G , we construct a CAT(0) cubical cell complexΣ as follows:For each i = 1 ,
2, let X i = R n i on which G i = Z n i acts geometrically by( a , . . . , a n i ) · ( x , . . . , x n i ) = ( x + a , . . . , x n i + a n i ) for any ( a , . . . , a n i ) ∈ G i and ( x , . . . , x n i ) ∈ X i . Here we consider that X i = R n i is the cubical cellcomplex whose 1-skeleton is the Cayley graph of G i = Z n i . Then we considerthe set { gX i | g ∈ G } , where each gX i is a copy of X i and gX i = hX i if andonly if g − h ∈ G i for each i = 1 ,
2. Let x i := 0 in X i = R n i . Then G i x i is alattice of X i = R n i . ere for each g ∈ G , we glue gX and gX by the one-point union gX ∨ gx = gx gX . Also we define the (max { n , n } )-dimensional cubical cell complex Σ asΣ = [ { gX | g ∈ G } ∪ [ { gX | g ∈ G } , where we identify gx = gx for any g ∈ G . Then Σ is contractible, since G = G ∗ G is the free-product of G and G . Moreover, Σ is a CAT(0)space on which G = G ∗ G naturally acts geometrically.Now we show that if the group G = G ∗ G acts geometrically on a CAT(0)space Y , then the actions of G on Σ and Y satisfy the condition ( ∗ ).Suppose that the group G = G ∗ G acts geometrically on a CAT(0)space Y . Then for each i = 1 ,
2, there exists y i ∈ Y such that the convex-hull C ( G i y i ) is isometric to R n i by [8, Theorem II.7.1]. Here we put y := y , Y := C ( G y ) and Y := C ( G y ). Then Y is isometric to R n on which G acts geometrically by lattice.We show that G acts cocompactly on Y . Let M = d Y ( y , C ( G y )).Then for any g, g ′ ∈ G , we have that d Y ( gy , C ( G y )) ≤ M and d Y ( g ′ y , C ( G y )) ≤ M . Hence [ gy , g ′ y ] ⊂ B ( C ( G y ) , M ), since Y is a CAT(0) space and C ( G y ) is convex. Also for any y, y ′ ∈ Y = C ( G y ), if d Y ( y, C ( G y )) ≤ M and d Y ( y ′ , C ( G y )) ≤ M then [ y, y ′ ] ⊂ B ( C ( G y ) , M ), since Y is a CAT(0) space and C ( G y ) is convex. Thuswe obtain that Y = C ( G y ) ⊂ B ( C ( G y ) , M ). Let N > G B ( y , N ) = Y . Then we note that B ( C ( G y ) , M ) ⊂ G B ( y , M + N ).Hence Y = C ( G y ) ⊂ B ( C ( G y ) , M ) ⊂ G B ( y , M + N ) . Here B ( y , M + N ) is compact. Thus the action of G on Y is cocompact.This implies that the group G acts geometrically on the CAT(0) space Y .By [8, Theorem II.7.1]. there exists y ′ ∈ Y such that the convex-hull C ( G y ′ ) ⊂ Y is isometric to R n . Let Y ′ := C ( G y ′ ).Since the action of G on Y is cocompact, there exists ¯ N > GB ( y , ¯ N ) = Y . Then we put ¯ Y = B ( Y , ¯ N ) and ¯ Y = B ( Y , ¯ N ). Here wenote that y ∈ Y ∩ Y and G ( ¯ Y ∩ ¯ Y ) = Y . Hence Y = [ { g ¯ Y | g ∈ G } and Y = [ { g ¯ Y | g ∈ G } . Here we show that the set ¯ Y ∩ ¯ Y is bounded. Indeed if ¯ Y ∩ ¯ Y isunbounded, then there exists α ∈ ∂ ( ¯ Y ∩ ¯ Y ). Then for i = 1 ,
2, there existsa sequence { g j y | j ∈ N } ⊂ G i y which converges to α in Y ∪ ∂Y . Hence α ∈ L ( G i ) for each i = 1 ,
2, where L ( G i ) = G i y ∩ ∂Y which is the limit setof G i in ∂Y . Then by [28, Theorem 4.15], α ∈ L ( G ) ∩ L ( G ) = L ( G ∩ G ) = L ( { } ) = ∅ , ecause G = G ∗ G is the free-product of G and G . This is a contradic-tion. Thus we obtain that ¯ Y ∩ ¯ Y is bounded.We also note that gG g − ∩ hG h − = { } for any g, h ∈ G and gG i g − ∩ hG i h − = { } for any g, h ∈ G such that gG i = hG i ( i = 1 , g ¯ Y ∩ h ¯ Y is bounded forany g, h ∈ G , and g ¯ Y i ∩ h ¯ Y i is bounded for any g, h ∈ G such that gG i = hG i ( i = 1 , S = { s ∈ G | B ( x , ¯ N ) ∩ B ( sx , ¯ N ) = ∅} − { } is a generating set of G . We can write S = { s , . . . , s k } ∪ { s ′ , . . . , s ′ l } where S = { s , . . . , s k } generates G and S = { s ′ , . . . , s ′ l } generates G . Here S ∩ S = ∅ , since G = G ∗ G .For any g ∈ G , we can write g = g h · · · g n h n for some g i ∈ G and h i ∈ G (where it may g = 1 or h n = 1). Then[ x , gx ] = [ x , g x ] ∪ [ g x , g h x ] ∪ · · · ∪ [ g h · · · g n x , gx ] , in Σ. By the structure of G and the argument above, we have that[ y , gy ] ∩ B ( g h · · · g i − h i − g i y , ¯ N ) = ∅ and[ y , gy ] ∩ B ( g h · · · g i h i y , ¯ N ) = ∅ for any i = 1 , . . . , n in Y .Hence, we obtain that the geometric action of G on Σ and any geometricaction of G on any CAT(0) space Y satisfy the condition ( ∗ ).Therefore, the group G = G ∗ G is an equivariant rigid CAT(0) group. Example 5.7.
By the same argument as above, we obtain that groups ofthe form G = Z n ∗ · · · ∗ Z n k ( n i ∈ N ) are equivariant rigid CAT(0) groups.Moreover, we can obtain that groups of the form G = Z n ∗ · · · ∗ Z n k ∗ A ∗ · · · ∗ A l (where n i ∈ N and each A j is a finite group) are equivariant rigid CAT(0)groups.6. Examples of non equivariant rigid CAT(0) groups
As an application of Theorem 1.2 and the condition ( ∗∗ ), we introducesome examples. Example 6.1.
Let G = F × Z , where F is the rank 2 free group generatedby { a, b } . Let T and T ′ be the Cayley graphs of F with respect to thegenerating set { a, b } such that(1) in T , all edges [ g, ga ] and [ g, gb ] ( g ∈ F ) have the unit length, and(2) in T ′ , the length of [ g, ga ] is 2 and the length of [ g, gb ] is 1 for any g ∈ F see Figure 1). Here we note that F acts naturally and geometrically on T and T ′ . T T ′ Figure 1.
Let X = T × R and Y = T ′ × R . We consider the natural actions of thegroup G on the CAT(0) spaces X and Y .Then the group G acts geometrically on the two CAT(0) spaces X and Y ,and the quasi-isometry gx gy (where x = (1 , ∈ X and y = (1 , ∈ Y ) does not extend continuously to any map from ∂X to ∂Y .Indeed, for example, we can consider the sequence { g n | n ∈ N } ⊂ F suchthat g = ab and g n = ( g n − a n − if n is even g n − b n − if n is oddfor n ≥
2. Here we note that the length of the words of g n in F is 2 n . Let¯ g n = ( g n , n ) ∈ F × Z for n ∈ N . Then the sequence { ¯ g n x } is a Cauchysequence in X ∪ ∂X and converges to the point [ α, π ] where α ∈ ∂T towhich { g n } converges. On the other hand, the sequence { ¯ g n y } is not aCauchy sequence in Y ∪ ∂Y . Thus by the condition ( ∗∗ ) and Theorem 1.2,the map φ : Gx → Gy ( gx gy ) does not continuously extend to anymap ¯ φ : ∂X → ∂Y .Here we note that the group G = F × Z is a rigid CAT(0) group whoseboundary is the suspension of the Cantor set. Example 6.2.
By the argument in Example 6.1, we obtain that everyCAT(0) group of the form G = F × H where F is a free group of rank n ≥ H is an infinite CAT(0) group, is non equivariant rigid.Indeed we can consider the Cayley graphs T and T ′ of F with respect tothe generating set { a , . . . , a n } of F such that(1) in T , all edges [ g, ga i ] ( g ∈ F , i = 1 , . . . , n ) have the unit length, and(2) in T ′ , the length of [ g, ga ] ( g ∈ F ) is 2 and the length of [ g, ga i ]( g ∈ F , i = 2 , . . . , n ) is 1. et Y be a CAT(0) space on which H acts geometrically and let X = T × Y and X ′ = T ′ × Y . Since H is an infinite CAT(0) group, there exists h ∈ H such that the order o ( h ) = ∞ ([50, Theorem 11]). Then h is a hyperbolicisometry of Y . Let σ be an axis for h in Y . Then, the natural actions of G = F × H on X and X ′ does not continuously extend to any map betweenboundaries ∂X and ∂X ′ by the same argument as Example 6.1. Indeed F × h h i ⊂ F × H = G , T × Im σ ⊂ X and T ′ × Im σ ⊂ X ′ .Here we note that ∂X = ∂T ∗ ∂Y and ∂X ′ = ∂T ′ ∗ ∂Y are homeomorphic.Also, if G = F × H acts geomretrically on a CAT(0) space X then theboundary ∂X is homeomorphic to ∂F ∗ ∂Z where Z is some CAT(0) spaceon which H acts geometrically by the splitting theorem in [35]. Hence G isrigid if and only if H is rigid.7. Remarks and questions
The author thinks that the main theorems, the conditions ( ∗ ) and ( ∗∗ )and some arguments in this paper could be used to investigate boundariesof CAT(0) groups and interesting open problems on(1) (equivariant) rigidity of boundaries of CAT(0) groups;(2) (equivariant) rigidity of boundaries of Coxeter groups;(3) (equivariant) rigidity of boundaries of Davis complexes of Coxetergroups;(4) (equivariant) rigidity of boundaries of CAT(0) spaces on which Cox-eter groups act geometrically as reflection groups;(5) (equivariant) rigidity of boundaries of CAT(0) spaces on which right-angled Coxeter groups act geometrically as reflection groups;(6) (equivariant) rigidity of boundaries of CAT(0) cubical complexes onwhich CAT(0) groups act geometrically;(7) (equivariant) rigidity of boundaries of CAT(0) cuboidal complexeson which CAT(0) groups act geometrically,etc.Here we can find some recent research on CAT(0) groups and their bound-aries in [11], [13], [22], [27], [35], [36], [39], [41], [43], [44], [48] and [51].Details of Coxeter groups and Coxeter systems are found in [6], [9] and [37],and details of Davis complexes which are CAT(0) spaces defined by Coxetersystems and their boundaries are found in [15], [16] and [47]. We can findsome recent research on boundaries of Coxeter groups in [10], [17], [18], [19],[33], [40]. Here we note that every cocompact discrete reflection group of ageodesic space becomes a Coxeter group [32], and if a Coxeter group W is acocompact discrete reflection group of a CAT(0) space X , then the CAT(0)space X has a structure similar to the Davis complex [34].The following theorem is known. Theorem 7.1 ([35, Theorem 5.2]) . If G and G are boundary rigid CAT(0)groups, then so is G × G . n research of (equivariant) boundary rigidity of CAT(0) groups, thefollowing natural open problem is important. Problem. If G and G are (equivariant) boundary rigid CAT(0) groups,then is G ∗ G also?We consider this problem for (4) above in Section 8 and we provide aconjecture on this problem for (7) above in Section 9.8. On equivariant rigid as reflection groups
A Coxeter group W is said to be equivariant rigid as a reflection group ,if for any two CAT(0) spaces X and Y on which W acts geometrically asa cocompact discrete reflection group of X and Y (cf. [32]), it obtain a W -equivariant homeomorphism of the boundaries ∂X and ∂Y as a continuousextension of the quasi-isometry φ : W x → W y defined by φ ( wx ) = wy ,where x ∈ X and y ∈ Y .Then we show the following. Theorem 8.1.
The following statements hold. (i)
If Coxeter groups W and W are equivariant rigid as reflectiongroups, then W ∗ W is also. (ii) For a Coxeter group W = W A ∗ W A ∩ B W B where W A ∩ B is finite,if W determines its Coxeter system up to isomorphism, and if theparabolic subgroups W A and W B are equivariant rigid as reflectiongroups then W is also.Proof. (i) Let W and W be Coxeter groups that are equivariant rigid asreflection groups and let W = W ∗ W . Suppose that the Coxeter group W acts geometrically on two CAT(0) spaces X and Y as cocompact discretereflection groups.Let ( W, S ) and (
W, S ′ ) be Coxeter systems obtained from the actions of W on X and Y as [32] respectively, and let C and D be chambers as W C = X and W D = Y .Then S and S ′ separate as the disjoint unions S = S ∪ S and S ′ = S ′ ∪ S ′ such that W S = W S ′ = W and W S = W S ′ = W , since W = W ∗ W .Let X i = W i C and Y i = W i D for i = 1 ,
2. Then each Coxeter group W i is cocompact discrete reflection group of X i and Y i (cf. [34]). Since W i is equivariant rigid as reflection groups, it obtain W i -equivariant homeo-morphism ¯ φ i : ∂X i → ∂Y i as a continuous extension of the quasi-isometry φ i : W i x → W i y defined by φ i ( wx ) = wy , where x ∈ C ⊂ X ∩ X and y ∈ D ⊂ Y ∩ Y .Let N = diam C and M = diam D . Then W B ( x , N ) = X and W B ( y , M ) = Y .Now we define a map ¯ φ : ∂X → ∂Y naturally as follows.Let α ∈ ∂X and let ξ α be the geodesic ray in X with ξ α (0) = x and ξ α ( ∞ ) = α . y [34, Theorem 3], there exists a sequence { s i | i ∈ N } ⊂ S such that each w n = s · · · s n is a reduced representation and d H (Im ξ α , P ) ≤ N , where d H is the Hausdorff distance and P = [ x , s x ] ∪ [ s x , s s x ] ∪ · · · ∪ [ s · · · s n − x , w n x ] ∪ · · · . Here since w n ∈ W = W ∗ W for any n ∈ N , we can write w n = a b · · · a k b k for some a i ∈ W and b i ∈ W .Then either(1) there exists n ∈ N such that w n ∈ w n W for any n ≥ n , that is, α ∈ w n ∂X ,(2) there exists n ∈ N such that w n ∈ w n W for any n ≥ n , that is, α ∈ w n ∂X , or(3) the sequences { a i } ⊂ W and { b i } ⊂ W are infinite, that is, α S { w ( ∂X ∪ ∂X ) | w ∈ W } .In the case (1), the sequence { w n s n +1 · · · s m y | m > n } converges tosome point ¯ α ∈ w n ∂Y , since W is equivariant rigid as reflection groups.Then we define ¯ φ ( α ) = ¯ α . (Here we note that ¯ φ ( α ) = w n ¯ φ ( β ) where β isthe point of ∂X to which the sequence { s n +1 · · · s m x | m > n } convergesin X ∪ ∂X .)Also in the case (2), similarly, we define ¯ φ ( α ) = ¯ α where ¯ α ∈ w n ∂Y isthe point to which the sequence { w n s n +1 · · · s m y | m > n } converges.In the case (3), s s s · · · · · · s n · · · · · · = a b a b · · · a k b k · · · · · · , where a i ∈ W and b i ∈ W . Here we note that X ∩ X = C and Y ∩ Y = D are bounded and compact. By [34, Theorem 3], P ∩ B ( a b · · · a k b k x , N ) = ∅ for any k ∈ N . Then the sequence { a b · · · a k b k y | k ∈ N } converges tosome point ¯ α ∈ ∂Y in Y ∪ ∂Y , because for each 1 < i < k , d ( a b · · · a i b i y , [ y , a b · · · a k b k y ]) ≤ M by [34, Theorem 3]. We define ¯ φ ( α ) = ¯ α .The map ¯ φ : ∂X → ∂Y defined above is well-defined, and by similararguments in sections above, we can show that ¯ φ is bijective, continuousand a W -equivariant homeomorphism.Therefore the Coxeter group W is equivariant rigid as a reflection group.(ii) Let W = W A ∗ W A ∩ B W B be a Coxeter group where W A ∩ B is finite.Suppose that W determines its Coxeter system up to isomorphism, theparabolic subgroups W A and W B are equivariant rigid as reflection groups,and the Coxeter group W acts geometrically on two CAT(0) spaces X and Y as cocompact discrete reflection groups.Then Coxeter systems ( W, S ) and (
W, S ′ ) are obtained from the actionsof W on X and Y as [32] respectively, and let C and D be chambers as C = X and W D = Y . Since W determines its Coxeter system up toisomorphism, two Coxeter systems ( W, S ) and (
W, S ′ ) are isomorphic.By [34], X A := W A C and X B := W B C are convex subspaces of X , Y A := W A D and Y B := W B D are convex subspaces of Y , and W A (resp. W B ) acts geometrically on the two CAT(0) spaces X A and Y A (resp. X B and Y B ) as cocompact discrete reflection groups.Since W A and W B are equivariant rigid as reflection groups, it ob-tain a W A -equivariant homeomorphism ¯ φ A : ∂X A → ∂Y A and a W B -equivariant homeomorphism ¯ φ B : ∂X B → ∂Y B as continuous extensionsof the quasi-isometries φ A : W A x → W A y defined by φ A ( wx ) = wy and φ B : W B x → W B y defined by φ B ( wx ) = wy respectively, where x ∈ C ⊂ X A ∩ X B and y ∈ D ⊂ Y A ∩ Y B respectively.Then we can define a W -equivariant homeomorphism ¯ φ : ∂X → ∂Y naturally as a similar construction to (i), since W A ∩ B is finite and X A ∩ X B = W A ∩ B C and Y A ∩ Y B = W A ∩ B D are bounded and compact. (cid:3) As an application of Theorem 8.1, we introduce some examples.
Example 8.2.
By Theorem 8.1 (i), any group of the form W = W ∗ · · · ∗ W n where each W i is a Gromov hyperbolic Coxeter group, an affine Coxetergroup or a finite Coxeter group, is equivariant rigid as a reflection group. Example 8.3.
By Theorem 8.1 (ii), any Coxeter group of the form W = ( · · · ( W A ∗ W B W A ) ∗ W B W A ) ∗ W B · · · ) ∗ W Bn − W A n where each W A i is a Gromov hyperbolic Coxeter group, an affine Coxetergroup or a finite Coxeter group, each W B i is finite and W determines itsCoxeter system up to isomorphism, is equivariant rigid as a reflection group. Example 8.4.
For example, by Theorem 8.1 (ii) and Example 8.3, theCoxeter groups defined by the diagrams in Figure 2 are equivariant rigidas reflection groups. Here these Coxeter groups determine their Coxetersystems up to isomorphism by [30] and [49]. • • •• • • ◗◗◗◗◗◗(cid:0)(cid:0)(cid:0) ❅❅❅ •• •••
Figure 2.
By similar arguments to Examples 6.1 and 6.2, we can construct nonequivariant rigid Coxeter groups (as reflection groups). xample 8.5. Let F = Z ∗ Z ∗ Z , let G = Z ∗ Z and let W = F × G .Then W is a non equivariant rigid Coxeter group (as a reflection group).Indeed we can consider the Cayley graphs T and T ′ of F with respect tothe generating set { a , a , a } of F such that(1) in T , all edges [ w, wa i ] ( w ∈ F , i = 1 , ,
3) have the unit length,(2) in T ′ , the length of [ w, wa ] ( w ∈ F ) is 2 and the length of [ w, wa i ]( w ∈ F , i = 2 ,
3) is 1.Let X = T × R and X ′ = T ′ × R . Then, the natural actions of W = F × G on X and X ′ does not continuously extend to any map between the boundaries ∂X and ∂X ′ by the same argument as Example 6.1.Here we note that the group W = F × G is a rigid CAT(0) group whoseboundary is the suspension of the Cantor set. Example 8.6.
By the arguments in Examples 6.1, 6.2 and 8.5, every Cox-eter group of the form W = F × G where F = Z ∗ · · · ∗ Z with rank n ≥ G is an infinite Coxeter group, is a non equivariant rigid (as a reflectiongroup).Here we note that if W = F × G acts geometrically on a CAT(0) space X then the boundary ∂X is homeomorphic to ∂F ∗ ∂Y where Y is someCAT(0) space on which G acts geometrically by the splitting theorem in[35]. Hence W is rigid if and only if G is rigid.9. Conjecture
Now we consider the CAT(0) group G = ( F × Z ) ∗ Z where F is the free group of rank 2.We note that F × Z is a rigid CAT(0) group and non equivariant rigid.Then the following conjecture arises. Conjecture.
The group G = ( F × Z ) ∗ Z will be a non-rigid CAT(0) groupwith uncountably many boundaries.This conjecture comes from the following idea.For p ≥ q ≥
1, let T p,q be the Cayley graph of the free group F with thegenerating set { a, b } such that • the length of [ g, ga ] is p and the length of [ g, gb ] is q for any g ∈ F .Then F × Z acts naturally on T p,q × R . By similar arguments to Ex-amples 5.3–5.7, we can construct a cuboidal cell complex Σ p,q on which G = ( F × Z ) ∗ Z acts geometrically, where the 1-skeleton of Σ p,q is theCayley graph of G and T p,q ⊂ Σ (1) p,q . hen from the argument of Example 6.1, the author expects that if pq = p ′ q ′ then the boundaries ∂ Σ p,q and ∂ Σ p ′ ,q ′ will be not homeomorphic.If this conjecture and this idea will be the case, then the right-angledArtin group ( F × Z ) ∗ Z , the right-angled Coxeter group( W A × W B ) ∗ W C where W A = Z ∗ Z ∗ Z , W B = Z ∗ Z and W C = Z , etc, will be alsonon-rigid CAT(0) groups with uncountably many boundaries.10. On rigidity
Let G and H be groups acting geometrically (i.e. properly and cocom-pactly by isometries) on metric spaces ( X, d X ) and ( Y, d Y ) respectively. Weconsider orbits Gx ⊂ X and Hy ⊂ Y where x ∈ X and y ∈ Y .Let φ : G → H be a map and let φ ′ : Gx → Hy ( gx φ ( g ) y ).Here if X and Y are Gromov hyperbolic spaces, CAT(0) spaces or Buse-mann spaces, then we can define the boundaries ∂X and ∂Y .Then it is well-known that if φ : G → H is an isomorphism then φ ′ : Gx → Hy is a quasi-isometry and moreover if G is Gromov hyperbolicthen φ ′ induces an equivariant homeomorphism ¯ φ : ∂X → ∂Y .Theorem 1.2 implies that if φ : G → H is an isomorphism and the map φ ′ : Gx → Hy satisfies the condition ( ∗∗ ) then φ ′ induces an equivarianthomeomorphism ¯ φ : ∂X → ∂Y . G · y X ⊃ Gx ←→ ∂X ↓ φ ↓ φ ′ ↓ ¯ φH · y Y ⊃ Hy ←→ ∂Y Then there are problems of rigidity.(I) If φ : G → H is an isomorphism then when does there exist anhomeomorphism ¯ φ : ∂X → ∂Y ?(II) If φ : G → H is an isomorphism then when does φ ′ induce anequivariant homeomorphism ¯ φ : ∂X → ∂Y ?(III) If X = Y and Gx = Hx then when are groups G and H virtuallyisomorphic (i.e. there exist finite-index subgroups G ′ and H ′ of G and H respectively such that G ′ and H ′ are isomorphic)?(IV) If X = Y and Gx = Hx then when do there exist finite-indexsubgroups G ′ and H ′ of G and H respectively such that G ′ and H ′ are conjugate in the isometry group Isom( X ) of X ?(V) If there is an isomorphism φ : G → H then when does there exist ahomeomorphism (or homotopy equivalence or strongly deformationretract) ψ : X/G → Y /H ? ere it seems that (III)–(V) are relate to [4], [20], [21], [26], [38], [41],[42], [45] and [46]. References [1] A. D. Alexandrov, V. N. Berestovskii and I. G. Nikolaev,
Generalized Riemannianspaces , Russ. Math. Surveys 41 (1986), 1–54.[2] W. Ballmann and M. Brin,
Orbihedra of nonpositive curvature , Inst. Hautes ´EtudesSci. Publ. Math. 82 (1995), 169–209.[3] W. Ballmann, M. Gromov and V. Schroeder,
Manifolds of Nonpositive Curvature ,Progr. Math. vol. 61, Birkh¨auser, Boston MA, 1985.[4] A. Bartels and W. L¨uck,
The Borel conjecture for hyperbolic and CAT(0)-groups ,Ann. of Math. (2) 175 (2012), 631–689[5] M. Bestvina,
Local homology properties of boundaries of groups , Michigan Math. J.43 (1996), 123–139.[6] N. Bourbaki,
Groupes et Algebr`es de Lie , Chapters IV-VI, Masson, Paris, 1981.[7] P. Bowers and K. Ruane,
Boundaries of nonpositively curved groups of the form G × Z n , Glasgow Math. J. 38 (1996), 177–189.[8] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature , Springer-Verlag, Berlin, 1999.[9] K. S. Brown,
Buildings , Springer-Verlag, 1980.[10] P. Caprace and K. Fujiwara,
Rank-one isometries of buildings and quasi-morphismsof Kac-Moody groups , Geom. Funct. Anal. 19 (2010), 1296-1319.[11] P. Caprace and M. Sageev,
Rank rigidity for CAT (0) cube complexes , Geom. Funct.Anal. 21 (2011), 851–891.[12] M. Coornaert and A. Papadopoulos,
Symbolic dynamics and hyperbolic groups , Lec-ture Notes in Math. 1539, Springer-Verlag, 1993.[13] C. B. Croke and B. Kleiner,
Spaces with nonpositive curvature and their ideal bound-aries , Topology 39 (2000), 549–556.[14] C. B. Croke and B. Kleiner,
The geodesic flow of a nonpositively curved graph man-ifold , Geom. Funct. Anal. 12 (2002), 479–545.[15] M. W. Davis,
Groups generated by reflections and aspherical manifolds not coveredby Euclidean space , Ann. of Math. 117 (1983), 293–324.[16] M. W. Davis,
Nonpositive curvature and reflection groups , in Handbook of geometrictopology (Edited by R. J. Daverman and R. B. Sher), pp. 373–422, North-Holland,Amsterdam, 2002.[17] A. N. Dranishnikov,
On boundaries of hyperbolic Coxeter groups , Topology Appl.110 (2001), 29–38.[18] A. N. Dranishnikov,
Boundaries of Coxeter groups and simplicial complexes withgiven links , J. Pure Appl. Algebra 137 (1999), 139–151.[19] H. Fischer,
Boundaries of right-angled Coxeter groups with manifold nerves , Topol-ogy 42 (2003), 423–446.[20] A. Furman,
Orbit equivalence rigidity , Ann. of Math. (2) 150 (1999), 1083–1108.[21] A. Furman,
Mostow-Margulis rigidity with locally compact targets , Geom. Funct.Anal. 11 (2001), 30–59.[22] R. Geoghegan and P. Ontaneda,
Boundaries of cocompact proper CAT(0) spaces ,Topology 46 (2007), 129–137.[23] E. Ghys and P. de la Harpe (ed),
Sur les Groupes Hyperboliques d’apr`es MikhaelGromov , Progr. Math. vol. 83, Birkh¨auser, Boston MA, 1990.[24] M. Gromov,
Hyperbolic groups , Essays in group theory (Edited by S. M. Gersten),pp. 75–263, M.S.R.I. Publ. 8, 1987.
25] M. Gromov,
Asymptotic invariants for infinite groups , Geometric Group Theory(G.A. Niblo and M.A. Roller, eds.), LMS Lecture Notes, vol. 182, Cambridge Uni-versity Press, Cambridge, 1993, pp. 1–295.[26] M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Geometric topology:recent developments (Montecatini Terme, 1990) (Berlin), Lecture Notes in Math.,vol. 1504, Springer, 1991, pp. 39–137.[27] U. Hamenst¨adt,
Rank-one isometries of proper CAT(0)-spaces , Discrete groupsand geometric structures; Workshop on Discrete Groups and Geometric Structures(Edited by K. Dekimpe, P. Igodt and A. Valette), AMS, 2009, pp. 43–60.[28] T. Hosaka,
Limit sets of geometrically finite groups acting on Busemann spaces ,Topology Appl. 122 (2002), 565–580.[29] T. Hosaka,
Parabolic subgroups of finite index in Coxeter groups , J. Pure Appl.Algebra 169 (2002), 215–227.[30] T. Hosaka,
Determination up to isomorphism of right-angled Coxeter systems , Proc.Japan Acad. Ser. A Math. Sci. 79 (2003), 33–35.[31] T. Hosaka,
The interior of the limit set of groups , Houston J. Math. 30 (2004),705–721.[32] T. Hosaka,
Reflection groups of geodesic spaces and Coxeter groups , Topology Appl.153 (2006), 1860–1866.[33] T. Hosaka,
On boundaries of Coxeter groups and topological fractal structures ,Tsukuba J. Math. 35 (2011), 153–160.[34] T. Hosaka,
Parabolic subgroups of Coxeter groups acting by reflections on CAT(0)spaces , Rocky Mount. J. Math. 42 (2012), 1207–1214.[35] T. Hosaka,
On splitting theorems for CAT(0) spaces and compact geodesic spaces ofnon-positive curvature , Math. Z. 272 (2012), 1037–1050.[36] G. C. Hruska,
Geometric invariants of spaces with isolated flats , Topology 44 (2005),441–458.[37] J. E. Humphreys,
Reflection groups and Coxeter groups , Cambridge UniversityPress, 1990.[38] Y. Kida,
Measure equivalence rigidity of the mapping class group , Ann. of Math. (2)171 (2010), 1851–1901.[39] M. Mihalik and K. Ruane,
CAT(0) groups with non-locally connected boundary , J.London Math. Soc. (2) 60 (1999), 757–770.[40] M. Mihalik, K. Ruane and S. Tschantz,
Local connectivity of right-angled Coxetergroup boundaries , J. Group Theory 10 (2007), 531–560.[41] N. Monod,
Superrigidity for irreducible lattices and geometric splitting , J. Amer.Math. Soc. 19 (2006), 781–814.[42] N. Monod and Y. Shalom,
Orbit equivalence rigidity and bounded cohomology , Ann.of Math. (2) 164 (2006), 825–878.[43] C. Mooney,
Examples of non-rigid CAT(0) groups from the category of knot groups ,Algebr. Geom. Topology 8 (2008), 1667–1690.[44] C. Mooney,
All CAT(0) boundaries of a group of the form H × K are CE equivalent ,Fund. Math. 203 (2009), 97–106.[45] G. D. Mostow, Quasi-conformal mappings in n -space and the rigidity of hyperbolicspace forms , Inst. Hautes Etudes Sci. Publ. Math. (1968), no. 34, 53–104.[46] G. D. Mostow, Strong rigidity of locally symmetric spaces , Princeton UniversityPress, Princeton, N.J., 1973, Annals of Mathematics Studies, no. 78.[47] G. Moussong,
Hyperbolic Coxeter groups , Ph.D. thesis, Ohio State University, 1988.[48] P. Papasoglu and E. L. Swenson,
Boundaries and JSJ decompositions of CAT(0)-groups , Geom. Funct. Analy. 19 (2009), 558–590.[49] D. Radcliffe,
Unique presentation of Coxeter groups and related groups , Ph.D. thesis,University of Wisconsin-Milwaukee, 2001.
50] E. L. Swenson,
A cut point theorem for CAT(0) groups , J. Differential Geom. 53(1999), 327–358.[51] J. M. Wilson,
A CAT(0) group with uncountably many distinct boundaries , J. GroupTheory 8 (2005), 229–238.[52] S. Yamagata,
On ideal boundaries of some Coxeter groups , Advanced Studies PureMath. 55 (2009), 345–352.
Department of Mathematics, Shizuoka University, Suruga-ku, Shizuoka 422-8529, Japan
E-mail address : [email protected]@ipc.shizuoka.ac.jp