Geometric invariants for locally compact groups

The features of a group being finitely generated or finitely presented are, respectively, the n=1 and n=2 cases of the finiteness property Fn. 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 Σn(G) (and their homological counterparts Σn(G;A), for A a ℤ[G]-module). Each set Σn(G) is a collection of homomorphisms G → R, refining property Fn, in the sense that G has type Fn precisely if Σn(G) contains the zero map. In the literature, Σ-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 Cn, which specialize to Fn for G discrete (though this fact is far from obvious at a first glance). In joint work with Kai-Uwe Bux and Elisa Hartmann, we have refined these properties Cn to sets Σn(G), with our definition recovering the classical Σ-sets in the discrete case. We have also generalized various results of classical Σ-theory to the setting of locally compact groups. In my talk, I will give an introduction to the theory of Σ sets and explain some of these results.

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There will be coffee and cake before the seminar in the common room