Geometry and Analysis on Groups Research Seminar

"Geometry and Analysis on Groups" Research Seminar

Time: 2014.12.02, 15:00--17:00
Location: Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "A combination theorem for cubulation in small cancellation theory over free products"
Speaker: Markus Steenbock
Abstract: We discuss a high-dimensional combination theorem to construct a CAT(0)--cubulation of hyperbolic groups given by small cancellation presentations over the free product out of the action of the free factors on CAT(0) cube complexes. To do so, we construct a space on which the groups act properly and cocompactly and which is built up from the CAT(0) cube complexes. We combine the walls on the CAT(0) cube complex to obtain a global wall structure on the space. The small cancellation condition over the free product can then be used to show that the group action gives an action on the walls that induces a proper and cocompact action on a CAT(0) cube complex. We obtain a new large class of CAT(0)--cubulable hyperbolic groups. These groups then have strong structural algebraic and geometric properties. Such groups satisfy Atiyah's conjecture on \(\ell^2\)-Betti numbers and in particular Kaplansky's zero-divisor conjecture.

The introductory talk covers the examples of wall-spaces that we use in the second part, CAT(0) cube complexes, polygonal complexes, and classical small cancellation conditions.

A. Martin is supported by the ERC-grant ANALYTIC 259527 of G. Arzhantseva. M. Steenbock is recipient of the DOC fellowship of the Austrian Academy of Sciences, and is partially supported by the ERC-grant ANALYTIC 259527 of G. Arzhantseva.