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Workshop

Geometric invariants for locally compact groups - the homotopical perspective

  • José Pedro Quintanilha
A3 01 (Sophus-Lie room)

Abstract

The features of a group being finitely generated or finitely presented are, respectively, the n=1 and n=2 cases of the finiteness property $F_n$. Towards the end of the last century, the question of when these finiteness conditions descend to subgroups of G led to the discovery of the sets $\Sigma^n(G)$ (and their homological counterparts $\Sigma^n(G;A)$, for A a Z[G]-module). Each set $\Sigma^n(G)$ is a collection of homomorphisms G --> R, refining property $F_n$ in the sense that G has type $F_n$ precisely if $\Sigma^n(G)$ contains the zero map. In the literature, $\Sigma$-sets are also often called BNSR-invariants, due to Bieri, Neumann, Strebel and Renz, who pioneered the theory.
Another direction in which to generalize finiteness properties is to consider groups G with a locally compact Hausdorff topology. In that setting, Abels and Tiemeyer introduced the compactness properties $C_n$, and used a well known criterion of Brown to show that they specialize to $F_n$ for G discrete. In joint work with Kai-Uwe Bux and Elisa Hartmann, we have refined these properties $C_n$ to sets $\Sigma^n(G)$, with our definition recovering the classical Sigma sets in the discrete case. We have also generalized various results of classical Sigma-theory to the setting of locally compact groups. I will give an introduction to the theory of Sigma sets and explain some of these results.

Antje Vandenberg

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Alexandra Linde

Augsburg University Contact via Mail

Christian Bär

Potsdam University

Bernhard Hanke

Augsburg University

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Burkhard Wilking

University of Münster