48 research outputs found
Infinite systolic groups are not torsion
We study -systolic complexes introduced by T. Januszkiewicz and J.
\'{S}wi\k{a}tkowski, which are simply connected simplicial complexes of
simplicial nonpositive curvature. Using techniques of filling diagrams we prove
that for the -skeleton of a -systolic complex is Gromov
hyperbolic. We give an elementary proof of the so-called Projection Lemma,
which implies contractibility of -systolic complexes. We also present a new
proof of the fact that an infinite group acting geometrically on a -systolic
complex is not torsion.Comment: Version 3, 27 pages, 10 figures. Major revision. Proof of Theorem 1.2
corrected, proof of Theorem 4.3 simplified, a reference to an alternative
proof of Theorem 7.4 added. Several definitions and lemmas adjusted and few
typos removed. Language and exposition improved. Version very similar to the
published versio
Coarse geometry of the fire retaining property and group splittings
Given a non-decreasing function we
define a single player game on (infinite) connected graphs that we call fire
retaining. If a graph admits a winning strategy for any initial
configuration (initial fire) then we say that has the -retaining
property; in this case if is a polynomial of degree , we say that
has the polynomial retaining property of degree .
We prove that having the polynomial retaining property of degree is a
quasi-isometry invariant in the class of uniformly locally finite connected
graphs. Henceforth, the retaining property defines a quasi-isometric invariant
of finitely generated groups. We prove that if a finitely generated group
splits over a quasi-isometrically embedded subgroup of polynomial growth of
degree , then has polynomial retaining property of degree . Some
connections to other work on quasi-isometry invariants of finitely generated
groups are discussed and some questions are raised.Comment: 16 pages, 1 figur
Cohomological and geometric invariants of simple complexes of groups
We investigate strictly developable simple complexes of groups with arbitrary
local groups, or equivalently, group actions admitting a strict fundamental
domain. We introduce a new method for computing the cohomology of such groups.
We also generalise Bestvina's construction to obtain a polyhedral complex
equivariantly homotopy equivalent to the standard development of the lowest
possible dimension.
As applications, for a group acting chamber transitively on a building of
type , we show that its Bredon cohomological dimension is equal to the
virtual cohomological dimension of and give a realisation of the building
of the lowest possible dimension.
We introduce the notion of a reflection-like action, and use it to give a new
family of counterexamples to the strong form of Brown's conjecture on the
equality of virtual cohomological dimension and Bredon cohomological dimension
for proper actions.
We show that the fundamental group of a simple complex of groups acts on
a tree with stabilisers generating a family of subgroups if and
only if its Bredon cohomological dimension with respect to is at
most one. This confirms a folklore conjecture under the assumption that a model
for the classifying space of for the family
has a strict fundamental domain.
In order to handle complexes of groups arising from arbitrary group actions,
we define a number of combinatorial invariants such as the block poset, which
may be of independent interest. We also derive a general formula for Bredon
cohomological dimension for a group admitting a cocompact model for
. As a consequence of both, we obtain a simple formula for
proper cohomological dimension of groups whose actions admit
a strict fundamental domain.Comment: 33 pages, 3 figures, 1 tabl
Commensurators of abelian subgroups in CAT(0) groups
We study the structure of the commensurator of a virtually abelian subgroup
in , where acts properly on a space . When
is a Hadamard manifold and is semisimple, we show that the commensurator of
coincides with the normalizer of a finite index subgroup of . When
is a cube complex or a thick Euclidean building and the
action of is cellular, we show that the commensurator of is an
ascending union of normalizers of finite index subgroups of . We explore
several special cases where the results can be strengthened and we discuss a
few examples showing the necessity of various assumptions. Finally, we present
some applications to the constructions of classifying spaces with virtually
abelian stabilizers.Comment: 20 page