Rigidity of pseudo-free group actions on contractible manifolds (Geometry of Transformation Groups and Combinatorics)

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(1)Title. Author(s). Citation. Issue Date. URL. Rigidity of pseudo-free group actions on contractible manifolds (Geometry of Transformation Groups and Combinatorics). KHAN, Qayum. 数理解析研究所講究録別冊 (2013), B39: 45-50. 2013-04. http://hdl.handle.net/2433/207824. Right. Type. Textversion. Departmental Bulletin Paper. publisher. Kyoto University.

(2) RIMS Kôkyûroku Bessatsu B39 (2013), 045−050. of. Rigidity. group actions. pseudo‐free. contractible manifolds. on. By. Qayum. In this. follows. article,. earlier. an. The author is. we. announcejoint. case. study. grateful. to the. of. Khan. *. work with Frank. pseudo‐free. organizers. Connolly. involutions. on. the. [6].. and Jim Davis. n ‐torus. This. carried out in. of the RIMS conferences where these results. [5].. were. disseminated in Asia:. Trransformation Groups and Surgery Theory (Masayuki Yamasaki, August 2010), Trransformation Groups and Combinatorics (Mikiya Masuda, June 2011). The Main Theorem. §1. Denition.. satisfies. Let \mathcal{F}\subset \mathrm{S} be families of. Property C_{\mathcal{F}\subset \mathrm{G}. says that $\Gamma$ satisfies. Below. M_{\mathcal{F}\subset \mathrm{G}. we. Property M_{\mathcal{F}\subset \mathrm{G}. group $\Gamma$. Furthermore,. .. consists of the trivial. the. finite‐by‐cyclic subgroups,. subgroup,. Let $\Gamma$ be. a. and. says that $\Gamma$ satisfies. one. self‐normalizing chain. increasing. {1}. \{1\}\subset. C(H). vc. fin \subset fbc \subset. consists of the. equipped. with. a. .. One. unique. Property NM_{\mathcal{F}\subset \mathrm{G}. vc. of. families,. subgroups,. where. fbc consists of. virtually cyclic subgroups.. \mathscr{S}( $\Gamma$). an. g. in. in $\Gamma$.. fin consists of the finite. group. We define. classes of contractible manifolds. We say that $\Gamma$. .. if every element H\in \mathrm{S}-\mathcal{F} is contained in. and each H_{\max} is. consider the. Denition.. a. if every element H\in g-\mathcal{F} has its centralizer. maximal element H_{\max} of G. if $\Gamma$ satisfies. subgroups of. as. the set of $\Gamma$ ‐homeomorphism. effective cocompact proper $\Gamma$ ‐action.. For any $\Gamma$ ‐space X , consider the free part of the action:. X_{free}. :=. { x\in X|. Our Main Theorem parameterizes. implies g=1\in $\Gamma$ }.. gx=x. \mathscr{S}( $\Gamma$). ,. and determines when it is. one. element.. Received March 21, 2012. Revised July 31, 2012. 2000 Mathematics Subject Classication(s): 57\mathrm{N}65, *. 57\mathrm{Q}12, 57\mathrm{R}67, 57\mathrm{S}30 Partially supported by the United States National Science Foundation (DMS‐0904276) Department of Mathematics, University of Notre Dame du Lac, Notre Dame IN 46556 U.S.A. \mathrm{e} ‐mail:. ©. qkhan@indiana.. edu. 2013 Research Institute for Mathematical. Sciences, Kyoto University.. All. rights. reserved..

(3) Theorem 1.1. (Main Theorem).. Property. 2. $\Gamma$ satises. Property \mathrm{M}_{\mathrm{f}\mathrm{b}\mathrm{c}\subset \mathrm{v}\mathrm{c} ,. 3. $\Gamma$ is. virtually torsion‐free. 4. there. exists. with. [X, $\Gamma$]\in \mathscr{S}( $\Gamma$). $\epsilon$:=(-1)^{n}. There is. .. a. (\mathrm{m}i\mathrm{d})( $\Gamma$). (\mathfrak{m}i\mathrm{d})( $\Gamma$). Conjecture. $\Gamma$ ‐homeomorphism. In. only. Smith. theory. proof,. The. following. was. vanishing. see. [7, 4]. of the. 2,. isomorphism of. we. locally. come. conelike. Wall realization:. .. of maximal innite. any. has. dihedral. subgroups of $\Gamma$.. representative with the. same. a. element. of. order two, then. \mathscr{S}( $\Gamma$). $\Gamma$ ‐manifold M , every $\Gamma$ ‐homotopy. $\Gamma$ ‐homeomorphism.. [6]. Notably,. for the. topological. actions. $\Gamma$ \mathrm{c}\mathrm{y}M,. points from Hypothesis (1), and Siebenmann. paragraph. locally. conelike from. Hypothesis (4).. of Theorem 1.1 is immediate from the. groups that. Let. occur as. n. be. an. the summands in. (1.1).. integer. Set $\epsilon$:=(-1)^{n}.. abelian groups:. \left{bginary}{l 0&ifn\equv0(mathr{}\ moathr{d}4)\ 0&ifnequv1(\mathr{} mo\athr{d}4)\ (mathb{Z}/2)^\inftyoplus(\mathb{Z}/4)^\infty& equiv2(\mathr{} mo\athr{d}4)\ (mathb{Z}/2)^\infty& equiv3(\mathr{} mo\athr{d}4). \enary}ight.. In Section. from. no. cocompact a. is achieved away from the. instructions. show that the above five. manifolds.. which cannot. ,. (ConnollyDavisRanicki).. construction, where gluing. In Section. 0\mapsto[X, $\Gamma$] given by. with. if $\Gamma$. or. Cappell. parameterization of (1.1). CAT(0). for. the full article. \mathrm{U}\mathrm{N}\mathrm{i}1_{n+ $\epsilon$}(\mathbb{Z};\mathbb{Z}, \mathbb{Z})\cong. on. a. is $\Gamma$ ‐homotopic to. result of the last. calculation. an. homotopy type of a finite complex,. in lower K ‐theory and in L ‐theory.. used to conclude the action must be. Then there is. handle. has. used to get isolated fixed. Theorem 1.2. The. classes. (mod4),. element. In this case,. theory was. of \mathscr{S}( $\Gamma$). n\equiv 0 , 1. equivalence f:M\rightarrow X For the. has the. type of links of singularities.. particular, if one. group. Assume:. UNil (; \mathbb{Z}, \mathbb{Z})\rightar ow^{\approx}\mathscr{S}( $\Gamma$). of conjugacy. is the set. each element. Furthermore,. X_{free}/ $\Gamma$. bijection of sets,. \oplus. (1.1) Here. a. n:=\mathrm{v}\mathrm{c}\mathrm{d}( $\Gamma$)>4,. where. 5. $\Gamma$ satises the Farrell‐Jones. Write. Let $\Gamma$ be. C_{\{\}\subset fin},. 1. $\Gamma$ satises. has. Khan. Qayum. 46. 3,. we. are. given by generalized Arf. properties. provide. singularities by. a. Riemannian manifold of. are. family. satisfied. of exotic. by. a. smooth. invariants.. certain actions. CAT(0) examples. nonpositive sectional. curvatures..

(4) Rigidity. 0F PsEuDo‐free group actions. §2.. Geometric consequences. A proper action $\Gamma$ \mathrm{c}\mathrm{y}X is. Denition.. X_{sing}:= {x\in X|. pseudo‐fr. gx=x for. 47. if the. ee. singular. set is discrete:. g\neq 1\in $\Gamma$ }.. some. originally established in [5] for the special case of the family crystallographic groups $\Gamma$=\mathbb{Z}^{n}\rangle\triangleleft {}_{-1}C_{2} for all n>3 More generally, we conclude: Theorem 1.1. was. of. .. Corollary. of. Euclidean space \mathrm{E}^{n}. Euclidean and. Corollary Suppose. $\Gamma$ is. on. isometries. a. like,. be viewed. as a. quotient. Let H be. X_{free}/ $\Gamma$. holds. a. By. C_{ $\Gamma$}(H)(\mathrm{y}X^{H}. .. class, CAT(0). bijection (1.1). [2,. of. Conjecture. Theorem \mathrm{A} ]. Both. [2,. for these groups,. holds. Since any two. a. Also,. lemma.. subgroup of. $\Gamma$. (see [3]):. spaces. dimension n>4.. rigidity. is. a. discrete. results. rely. Theorem \mathrm{B} ].. points. in X. since $\Gamma$(\mathrm{y}X is. arejoined. locally. .. [2], Hypothesis (5). Since the action $\Gamma$ \mathrm{c}\mathrm{y}X is. cone‐. unique. Hypothesis (1). D ‐invariant. holds.. pseudo‐free,. Note the proper action $\Gamma$ \mathrm{c}\mathrm{y}X restricts to. .. \square. compact topological \partial ‐manifold. Hence. is finite. Therefore. So. holds.. holds.. recent theorem of BartelsLück. C(H) D\in $\nu$ \mathrm{c}( $\Gamma$)-\mathrm{f}\mathrm{b}\mathrm{c}( $\Gamma$) There. a. proper. holds.. geodesic. line. \ell_{D}\subset X,. Hypothesis (1), see [6], that D is isomorphic to the infinite D_{\infty}=C_{2}*C_{2} Write D=\langle a, b|a^{2}=b^{2}=1\rangle Let x and y be the. follows. It follows from. dihedral group,. unique fixed points. joined by a. a. .. in X of. unique geodesic. closed subset of X. closed subset of \ell\approx \mathbb{R}. geodesic.. a. .. .. and b. .. Since ab has infinite. segment $\sigma$\subset X. Suppose. .. \ell\subset X is. Hence D $\sigma$=\ell. It remains to prove such is. Then the. single point.. a. isometries. locally conelike, pseudo‐free, cocompact. is the interior of. a. Selberg’s. CAT(0) topological manifold of. ee,. of. group. bijection (1.1). The. .. 2.2 below and. broader. a. X is contractible.. nontrivial finite. the fixed set X^{H} is. Let. Corollary. By assumption, Hypothesis (3). Hypothesis (4). action. space \mathbb{H}^{n} with n>4. generalization. unique geodesic segment, the. a. of X.. the truth of the FarrellJones. by. a. Let X be. cocompact, discrete. ee,. spaces fit into. virtually torsion‐fr. a. can. Proof.. is. hyperbolic. hyperbolic. 2.2.. of. proper group. This. or. pseudo‐fr. a. This will be immediate from. Proof.. as. Let $\Gamma$ be. 2.1.. an. .. a. .. Note that D $\sigma$ is. D‐invariant. Therefore,. action of b is. $\tau$. and y. are. distinct, to \mathbb{R} and. line. Then D $\sigma$\approx \mathbb{R} is. any such \ell is. unique.. \ell exists, that is, the D‐invariant embedded line D $\sigma$\subset X. unique geodesic segment joining. isometric,. x. homeomorphic. geodesic. It suffices to show that the segment $\sigma$\cup b $\sigma$\subset X. Let $\tau$\subset X be the. order,. is b ‐invariant and its. x. midpoint. joining. and bx. m,. x. and bx is. geodesic.. Since b^{2}=1 and the. with respect to the. arclength.

(5) Khan. Qayum. 48. is fixed. parameterization, \ell_{D}:=D $\sigma$. is the. unique. b. by. Hence y=m and. .. D ‐invariant. ,. has. .. is the. $\sigma$\cup b $\sigma$= $\tau$ is. geodesic line in X. D\subseteq D' then \ell_{D'} is. D'\in $\nu$ \mathrm{c}( $\Gamma$)-\mathrm{f}\mathrm{b}\mathrm{c}( $\Gamma$) satisfies D'\subseteq \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}_{ $\Gamma$}(\el _{D}) Therefore, since \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}_{ $\Gamma$}(\el _{D}) If. so. unique maximal virtually cyclic subgroup of. holds. Now. Geometric. §3.. natural. CAT(0) examples. such. Thanks go to Mike Davis for feedback Let K be. (X, $\Gamma$). of. for such infinite $\Gamma$ with 2‐torsion. source. an. Davis constructs. containing. abstract a. on. this. simplicial complex. cubical cell. Thus. .. Herein,. Hypothesis (2) \square. \mathrm{A}. exist which cannot be Riemannian.. reflection groups of. are. and. exposition. a. guide. with finite vertex set S. .. right‐angled Coxeter. convex. polytopes.. to define $\Gamma$ below.. [8,. 1.2], system (W_{K}, S) :. In. Section. we use. the set‐theoretic notation. B^{A}:= {functions f. :. A\rightarrow B }.. The link of each vertex of P_{K} , hence of each vertex of the universal. isomorphic. to the. isometric action. \dot{L} From these. geometric. realization. W_{K}(\mathrm{y}\overline{P_{K}. covering. actions, Davis obtains. an. |K|\subset[0. ,. 1. ]^{S}. There is. .. a. cocompact,. the natural reflection action W_{K}(\mathrm{y}. identification and. an. \overline{P_{K}. cover. ,. is. proper,. [1, 1] S.. exact sequence of groups:. 1 \rightarrow$\pi$_{1}(P_{K})=[W_{K}, W_{K}]\rightarrow W_{K}\rightarrow^{ $\varphi$}\{-1, 1\}^{S}\rightarrow 1.. (3.3) The all. barycentric. linearly. whenever. a. Since bK is. Then,. since. call the. subdivision bK is the abstract. ordered subsets of K of. finite subset of vertices. Let K be. right‐angled Coxeter. cardinality n+1. .. A. whose. n. ‐simplices. simplicial complex. pairwise joined by edges, they. are. an. abstract. simplicial complex. group W and the cubical. virtually torsion‐free subgroup $\Gamma$\underline{\triangleleft}W Proof.. simplicial complex. span. a. flag if, simplex.. is. flag, by [8, Proposition 1.2.3], the induced metric on X:=P_{bK} is CAT(0). Pb_{bK} is aspherical, (3.3) implies that W:=W_{bK} is virtually torsion‐free.. Lemma 3.1.. a. it. W_{K} = \langle S \{s^{2}=1\}_{s\in S}, \{[s, t]=1\}_{\{s,t\}\in K}\rangle.. (3.2). is. ,. P_{K} := \displaystyle \bigcup_{ $\sigma$\in K}[-1, 1]^{ $\sigma$}\times\{-1, 1\}^{S- $\sigma$}\subset[-1, 1]^{ S}. (3.1). are. \ell_{D}\approx \mathbb{R}. examples. and. complex P_{K}. D. on. so. ,. bijection (1.1).. Theorem 1.1 in order to obtain the. apply. Indeed,. $\Gamma$. \ell_{D'}=\ell_{D}. D ‐invariant, hence. proper isometric action. a. Therefore. geodesic.. Note bK has vertex set K. .. with. complex. X. finite. vertex set. Re‐. dened. above.. with torsion such that $\Gamma$ \mathrm{c}\mathrm{y}X is. Write n:=\dim K. .. Consider the. $\theta$:\{-1, 1\}^{K}\rightar ow\{-1, 1\}^{n+1} f\displaystyle\mapsto(\prod_{\dim$\sigma$=i}f($\sigma$) _{i=0}^{n} ;. There. pseudo‐fr. ee.. epimorphism.

(6) Rigidity. Define. normal. a. 49. 0F PsEuDo‐free group actions. subgroup. $\Gamma$:=( $\theta$\circ $\varphi$)^{-1}\langle(-1, \ldots, -1)\rangle\underline{\triangleleft}W. Note $\Gamma$ is. fact, (3.3). torsion‐free: in. virtually. restricts to. an. exact sequence. 1 \rightarrow[W, W]\rightarrow $\Gamma$\rightarrow^{ $\varphi$}$\theta$^{-1}\langle(-1, \ldots, -1)\rangle\rightarrow 1. W_{\triangle^{n}}=\{-1, 1\}^{n+1}\mathrm{c}\sim[-1, 1]^{n+1}=P_{\triangle^{n}}. Observe the reflection action. \langle(-1, \ldots, -1)\rangle(\mathrm{y}P_{\triangle^{n}}. pseudo‐free. action. induced. P‐constructions. respect. on. by. inclusion and. an. homomorphism $\theta$\circ $\varphi$. to the. Then W\mathrm{c}\mathrm{y}P_{bK} restricts to is W ‐equivariant and is. a. :. on. a. cubical map. a. projection,. action. each. on. equivariant with. each cube. $\Gamma$ \mathrm{c}\mathrm{y}P_{bK} So, since the .. cube, the. a. Pb_{bK}\rightarrow P_{\triangle^{K}}\rightarrow P_{\triangle^{n}},. that is. W\rightarrow W_{\triangle^{n}} and is injective. pseudo‐free. injective. There is. .. restricts to. action $\Gamma$ \mathrm{c}\mathrm{y}X is. [ 1, 1 ]^{ $\sigma$}.. X\rightarrow P_{bK}. map. pseudo‐free.. \square. specify the exotic CAT(0) examples W\mathrm{c}\mathrm{y}X of DavisJanuskiewicz, recounted in [8, Example 10.5.3]. The key feature is that X is a topological manifold of any given dimension n\geq 7 but it not simply connected at infinity. Hence X is a contractible n ‐dimensional manifold, not homeomorphic to \mathbb{R}^{n}. Example. Now. 3.2.. we. proceed. to. ,. Let 3\leq m\leq n-4. $\pi$\neq 1 integral homology. damental group same. 3‐sphere.. .. Start with. (Recall. .. groups. Write C for the. as. a. a. triangulated homology. homology. S^{m}. ) For. complement of. m. A. .. Thicken C into. :=C\times D^{n-m-1}. with. a. M. example,. is. m,. a. can. the open star of. compact, triangulated \partial ‐manifold of dimension. \partial C\approx S^{m-1}. ‐sphere. a. m. M with fun‐. ‐sphere. closed manifold with the be the Poincaré vertex in M. .. homology. Then C is. with the fundamental group. $\pi$ ,. a. and. compact, triangulated \partial ‐manifold. \displaystyle \partial A\approx(C\times S^{n-m-2})\bigcup_{(S^{m-1}\times S^{n-m-2})}(S^{m-1}\times D^{n-m-1}). .. $\pi$_{1}(\partial A)\rightarrow$\pi$_{1}(A)= $\pi$ of fundamental groups is an isomorphism. homology (n-2) ‐sphere, since M is a homology m ‐sphere.. Note the induced map. Furthermore, \partial A Define. a. is. simply. a. connected. homology‐manifold L. Observe that L is not. a. :=A\displaystyle \bigcup_{\partial A} Cone (\partial A). theorem of. Nonetheless, by More. this is true for any. nected links. Davis. complex. X,. 2.2 calculates. cone. point. suspension of L is. a. c. simplicial complex of. right‐angled Coxeter system (W, S). ,. and. L. .. is not. topological. triangulated homology‐manifold. Write K for the abstract. Each vertex link is. lary. Edwards,. the. by. .. manifold since the link of the. a. generally,. L of dimension n-1. with. a. sphere.. manifold.. simply. con‐. Consider the cubical. subgroup. $\Gamma$ from Lemma 3.1.. |bK|\approx|K|=L Thus X is a topological manifold. Therefore, Corol‐ \mathscr{S}( $\Gamma$) But, by [8, Theorem 9.2.2], $\pi$_{1}(L-c)\neq 1 implies $\pi$_{1}^{\infty}(X)\neq 1. .. ..

(7) Qayum. 50. Finally, reliance. Khan. the axiomatic formulation of Theorem 1.1 is. on convex. axioms; thanks. Example. geometry in the proof. Here is. Speyer. go to David. 3.3.. for. For any commutative. a non‐convex. it out. pointing. worthwhile;. ring. on. example. removes. the. to illustrate the. http://mathoverflow.net.. R , recall the R ‐Heisenberg group. \mathrm{H}\mathrm{e}\mathrm{i}(R):=\{ left(\begin{ar ay}{l 1x&z\ 01&y\ 0 1& \end{ar ay}\right)x,y z\inR\} subsetGL(3,R). .. $\omega$:=\exp(2 $\pi$ i/3)\in \mathbb{C} is a primitive third root of unity. Also consider the diagonal matrix D:= diag ( 1, $\omega,\ \omega$^{2})\in GL(3, \mathbb{C}) Define a semidirect product $\Gamma$=\mathrm{H}\mathrm{e}\mathrm{i}(\mathbb{Z}[ $\omega$])\rangle\triangleleft C_{3} where the C_{3} ‐action is given by conju‐ gation by D in GL(3, \mathbb{C}) Take X=\mathrm{H}\mathrm{e}\mathrm{i}(\mathbb{C}) Then $\Gamma$ satisfies Hypotheses (15), using a theorem of BartelsFarrellLück [1]. Therefore: \mathscr{S}( $\Gamma$)=\{[X, $\Gamma$]\} by Theorem 1.1. Recall the Solvable Subgroup Theorem [3, II.7.8]: if a virtually solvable group $\Gamma$ admits a cocompact proper action by isometries on a CAT(0) space, then $\Gamma$ must be Consider the Eisenstein. integers \mathb {Z}[ $\omega$]. it. where. ,. .. ,. .. .. ,. virtually. abelian.. Therefore. our. However,. $\Gamma$ cannot act. our. virtually. group $\Gamma$ is. cocompactly. and. solvable but not. properly by. isometries. virtually. on a. abelian.. CAT(0). space.. References. [1]. A.. Bartels, F. T. Farrell, and W. Lück. The Farrell‐Jones Conjecture for cocompact lattices virtually connected Lie groups. http://arxiv.org/abs/1101.0469. Arthur Bartels and Wolfgang Lück. The Borel Conjecture for hyperbolic and CAT(0)‐ groups. Ann. of Math. (2), 175(2):631-689 2012. Martin R. Bridson and André Haeiger. Metric spaces of non‐positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaft en [Fundamental Principles of Math‐ ematical Sciences]. Springer‐Verlag, Berlin, 1999. Frank Connolly and James F. Davis. The surgery obstruction groups of the innite dihedral group. Geom. Topol., 8:1043−1078 (electronic), 2004. Frank Connolly, James F. Davis, and Qayum Khan. Topological rigidity and H_{1} ‐negative in. [2]. ,. [3]. [4] [5]. involutions. http://arxiv.org/abs/1102.2660. Davis, and Qayum Khan. Topological rigidity and actions on Connolly, [6] contractible manifolds with discrete singular set. In preparation. [7] Frank Connolly and Andrew Ranicki. On the calculation of UNil. Adv. Math., 195(1):205on. Frank. tori.. James F.. 258, 2005.. [8]. Michael W. Davis.. Mathematical. The geometry and. Society Monographs. topology of Coxeter groups, volume 32 of London University Press, Princeton, NJ, 2008.. Series. Princeton.

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