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Dismantlability of weakly systolic complexes and applications. (English) Zbl 1376.20047
Summary: The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups.
The fixed point theorem is proved by using a graph-theoretical tool – dismantlability. In particular we show that $$1$$-skeleta of weakly systolic complexes, i.e., weakly bridged graphs, are dismantlable. On the way we show numerous characterizations of weakly bridged graphs and weakly systolic complexes.

##### MSC:
 20F65 Geometric group theory 57M07 Topological methods in group theory 57M15 Relations of low-dimensional topology with graph theory 20F67 Hyperbolic groups and nonpositively curved groups 05C12 Distance in graphs
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