Damian Osajda

Dismantlability of weakly systolic complexes and applications

In this paper, we investigate the structural properties of weakly systolic complexes introduced recently by the second author and of their 1-skeletons, the weakly bridged graphs. We present several characterizations of weakly systolic complexes and weakly bridged graphs. Then we prove that weakly bridged graphs are dismantlable. Using this, we establish the fixed point theorem for weakly systolic complexes. As a consequence, we get results about conjugacy classes of finite subgroups and classifying spaces for finite subgroups of weakly systolic groups. As immediate corollaries, we obtain new results on systolic complexes and

Realisation and dismantlability

We prove that a finite group acting on an infinite graph with dismantling properties fixes a clique. We prove that in the flag complex spanned on such a graph the fixed point set is contractible. We study dismantling properties of the arc, disc and sphere graphs. We apply our theory to prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out.Fn/ fixes a filling (respectively simple) clique in the appropriate graph. We deduce some realisation theorems, in particular the Nielsen realisation problem in the case of a nonempty set of punctures. We also prove that infinite H have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and Out.Fn/. 20F65

A construction of hyperbolic Coxeter groups

We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.

On asymptotically hereditarily aspherical groups

We undertake a systematic study of asymptotically hereditarily aspherical (AHA) groups, the class of groups introduced by Tadeusz Januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups. We provide many new examples of AHA groups, also in high dimensions. We relate the AHA property with the topology at infinity of a group, and deduce in this way some new properties of (weakly) systolic groups. We also exhibit an interesting property of boundaries at infinity for a few classes of AHA groups.

Connectedness at infinity of systolic complexes and groups

By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by euclidean space of dimension at least three. We also study semistability at infinity for some systolic groups.

On two conjectures of Maurer concerning basis graphs of matroids

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14 (1973), 216-240). We also establish Conjecture 1 from the same paper about the redundancy of the conditions in the characterization of basis graphs. We indicate positive-curvature-like aspects of the local properties of the studied complexes. We characterize similarly the corresponding 2-dimensional complexes of even

Residually finite non-exact groups

\Delta

Group cubization (with an appendix by Mikael Pichot)

-matroids.

Small cancellation labellings of some infinite graphs and applications

We construct the first examples of residually finite non-exact groups.

Boundaries of right-angled hyperbolic buildings

We present a procedure of group cubization: It results in a group whose some features resemble the ones of a given group, and which acts without fixed points on a CAT(0) cubical complex. As a main application we establish lack of Kazhdans property (T) for Burnside groups.