The geometric invariants \(\Sigma^*\) of a group refine finiteness properties: a point in \(\Sigma^{1}(G)\) can be thought of as a direction toward infinity along which the group \(G\) is finitely generated, a point in \(\Sigma^{2}(G)\) corresponds to a direction toward infinity in which \(G \) is finitely presented, and so on for higher invariants. Our main objectives concern the geometric invariants of arithmetic groups. Since arithmetic groups are lattices in locally compact groups we also want to develop the theory of higher \(\Sigma\)-invariants of locally compact groups (recently discovered by Schick and Bux) and elucidate the relationship of the geometric invariants of lattices and their ambient locally compact groups.
Publications
We develop an algorithm for recognizing whether a character belongs to \(\Sigma^m\). In order to apply it we just need to know that the ambient group is of type \(\mathrm{FP}_m\) or of type \( \mathrm{F}_2\) and that the word problem is solvable for this group. Then finite data is sufficient proof of membership in \(\Sigma^m\), not just for the given character but also for a neighborhood of it.
Related project(s):
39Geometric invariants of discrete and locally compact groups
In this paper we develop the theory of homological geometric invariants (following Bieri-Neumann-Strebel-Renz) for locally compact Hausdorff groups. The homotopical version is treated elsewhere. Both versions are connected by a Hurewicz-like theorem.
Related project(s):
39Geometric invariants of discrete and locally compact groups
We extend the classical theory of homotopical Σ-sets, defined by Bieri, Neumann, Renz and Strebel for abstract groups, to locally compact Hausdorff groups. Given such a group G, our geometric invariants are sets of continuous homomorphisms G→R ("characters"). They match the classical Σ-sets if G is discrete, and refine the homotopical compactness properties of Abels and Tiemeyer. Moreover, our theory recovers the definition of low-dimensional geometric invariants for topological gropus proposed by Kochloukova.
Related project(s):
39Geometric invariants of discrete and locally compact groups
Team Members
Prof. Dr. Kai-Uwe Bux
Project leader
Universität Bielefeld
kaiuwe.bux(at)gmail.com
Dr. Dorian Chanfi
Researcher
JLU Gießen
dorian.chanfi(at)math.uni-giessen.de
Dr. Elisa Hartmann
Researcher
Universität Bielefeld
elisa.hartmann(at)math.uni-bielefeld.de
Dr. José Pedro Quintanilha
Researcher
Universität Heidelberg
jquintanilha(at)mathi.uni-heidelberg.de
Prof. Dr. Stefan Witzel
Project leader
JLU Gießen
stefan.witzel(at)math.uni-giessen.de