×

zbMATH — the first resource for mathematics

Boundaries of systolic groups. (English) Zbl 1271.20056
From the introduction: For all systolic groups we construct boundaries which are \(EZ\)-structures. This implies the Novikov conjecture for torsion-free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.
There are many notions of boundaries of groups used for various purposes. In this paper we focus on the notions of \(Z\)-structure and \(EZ\)-structure introduced by M. Bestvina [Mich. Math. J. 43, No. 1, 123-139 (1996; Zbl 0872.57005)] and studied, e.g., by A. N. Dranishnikov [Contemp. Math. 394, 77-85 (2006; Zbl 1106.20034)] and F. T. Farrell and J.-F. Lafont [Comment. Math. Helv. 80, No. 1, 103-121 (2005; Zbl 1094.57003)].
Our main result is the following. Theorem A (Theorem 6.3) Let a group \(G\) act geometrically by simplicial automorphisms on a systolic complex \(X\). Then there exists a compactification \(\overline X=X\cup\partial X\) of \(X\) satisfying the following: (1) \(\overline X\) is a Euclidean retract (ER). (2) \(\partial X\) is a \(Z\)-set in \(\overline X\). (3) For every compact set \(K\subset X\), \((gK)_{g\in G}\) is a null sequence. (4) The action of \(G\) on \(X\) extends to an action by homeomorphisms of \(G\) on \(\overline X\).
A group \(G\) as in Theorem A is called a systolic group. It is a group acting geometrically (i.e., cocompactly and properly discontinuously) by simplicial automorphisms on a systolic complex – contractible simplicial complex satisfying some local combinatorial conditions. Systolic complexes were introduced by V. Chepoi [Adv. Appl. Math. 24, No. 2, 125-179 (2000; Zbl 1019.57001)] (under the name of bridged complexes) and, independently, by T. Januszkiewicz and J. Świątkowski [Publ. Math., Inst. Hautes Étud. Sci. 104, 1-85 (2006; Zbl 1143.53039)] and by F. Haglund [Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71 (2003)] (in Section 2 we give some background on them). Systolic complexes (groups) have many properties of nonpositively curved spaces (groups). There are systolic complexes that are not CAT(0) when equipped with the path metric in which every simplex is isometric to the standard Euclidean simplex. On the other hand, there are systolic groups that are not hyperbolic, e.g., \(\mathbb Z^2\). Summarizing, the systolic setting does not reduce to the CAT(0) or to the hyperbolic one.
Here we give the other definitions that appear in the statement of Theorem A. A compact space is a Euclidean retract if it can be embedded in some Euclidean space as its retract. A closed subset \(Z\) of a Euclidean retract \(Y\) is called a \(Z\)-set if for every open set \(U\subset Y\), the inclusion \(U\setminus Z\hookrightarrow U\) is a homotopy equivalence. A sequence \((K_i)_{i=1}^\infty\) of subsets of a topological space \(Y\) is called a null sequence if for every open cover \(\mathcal U=\{U_i\}_{i\in I}\) of \(Y\) all but finitely many \(K_i\) are \(\mathcal U\)-small, i.e., for all but finitely many \(j\) there exist \(i(j)\) such that \(K_j\subset U_{i(j)}\).
Conditions (1), (2) and (3) of Theorem A mean (following M. Bestvina, [loc. cit.], where only free actions were considered, and A. N. Dranishnikov, [loc. cit.]) that any systolic group \(G\) admits a \(Z\)-structure \((\overline X,\partial X)\). The notion of an \(EZ\)-structure, i.e., a \(Z\)-structure with the additional property (4), was explored by F. T. Farrell and J.-F. Lafont [loc. cit.] (in the case of a free action).

MSC:
20F65 Geometric group theory
57M07 Topological methods in group theory
20F69 Asymptotic properties of groups
20F67 Hyperbolic groups and nonpositively curved groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] G Arzhantseva, M R Bridson, T Januszkiewicz, I J Leary, A Minasyan, J Świ\catkowski, Infinite groups with fixed point properties, Geom. Topol. 13 (2009) 1229 · Zbl 1197.20034 · doi:10.2140/gt.2009.13.1229 · arxiv:0711.4238
[2] A Bartels, W Lück, The Borel Conjecture for hyperbolic and \(\mathrm{CAT}(0)\)-groups · Zbl 1256.57021 · arxiv:0901.0442v1
[3] M Bestvina, Local homology properties of boundaries of groups, Michigan Math. J. 43 (1996) 123 · Zbl 0872.57005 · doi:10.1307/mmj/1029005393
[4] M Bestvina, G Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469 · Zbl 0767.20014 · doi:10.2307/2939264
[5] B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145 · Zbl 0911.57001 · doi:10.1007/BF02392898
[6] M R Bridson, D T Wise, \(\mathcal{VH}\) complexes, towers and subgroups of \(F\times F\), Math. Proc. Cambridge Philos. Soc. 126 (1999) 481 · Zbl 0942.20009 · doi:10.1017/S0305004199003503
[7] G Carlsson, E K Pedersen, Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory, Topology 34 (1995) 731 · Zbl 0838.55004 · doi:10.1016/0040-9383(94)00033-H
[8] V Chepoi, Graphs of some \(\mathrm{CAT}(0)\) complexes, Adv. in Appl. Math. 24 (2000) 125 · Zbl 1019.57001 · doi:10.1006/aama.1999.0677
[9] V Chepoi, D Osajda, Dismantlability of weakly systolic complexes and applications, in preparation · Zbl 1376.20047
[10] C B Croke, B Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549 · Zbl 0959.53014 · doi:10.1016/S0040-9383(99)00016-6
[11] F Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proc. London Math. Soc. \((3)\) 86 (2003) 666 · Zbl 1031.20039 · doi:10.1112/S0024611502013989
[12] A N Dranishnikov, On Bestvina-Mess formula (editors R Grigorchuk, M Mihalik, M Sapir, Z \vSunik), Contemp. Math. 394, Amer. Math. Soc. (2006) 77 · Zbl 1106.20034
[13] J Dugundji, Topology, Allyn and Bacon (1966) · Zbl 0144.21501
[14] T Elsner, Systolic spaces with isolated flats, submitted
[15] T Elsner, Flats and the flat torus theorem in systolic spaces, Geom. Topol. 13 (2009) 661 · Zbl 1228.20033 · doi:10.2140/gt.2009.13.661
[16] F T Farrell, J F Lafont, EZ-structures and topological applications, Comment. Math. Helv. 80 (2005) 103 · Zbl 1094.57003 · doi:10.4171/CMH/7
[17] F Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71 (2003)
[18] F Haglund, J Świ\catkowski, Separating quasi-convex subgroups in \(7\)-systolic groups, Groups Geom. Dyn. 2 (2008) 223 · Zbl 1147.20040 · doi:10.4171/GGD/37 · www.ems-ph.org
[19] T Januszkiewicz, J Świ\catkowski, Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. (2006) 1 · Zbl 1143.53039 · doi:10.1007/s10240-006-0038-5 · numdam:PMIHES_2006__104__1_0 · eudml:104219
[20] T Januszkiewicz, J Świ\catkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007) 727 · Zbl 1188.20043 · doi:10.2140/gt.2007.11.727
[21] W Lück, Survey on classifying spaces for families of subgroups (editors L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk), Progr. Math. 248, Birkhäuser (2005) 269 · Zbl 1117.55013
[22] P Papasoglu, E Swenson, Boundaries and JSJ decompositions of \(\mathrm{CAT}(0)\)-groups · Zbl 1226.20038 · doi:10.1007/s00039-009-0012-8
[23] P Przytycki, \(\underline EG\) for systolic groups, Comment. Math. Helv. 84 (2009) 159 · Zbl 1229.20036 · doi:10.4171/CMH/156 · www.ems-ph.org
[24] D Rosenthal, Split injectivity of the Baum-Connes assembly map · Zbl 1286.19003 · doi:10.4310/PAMQ.2012.v8.n2.a4 · arxiv:math/0312047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.