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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
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