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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, which specialize to F_n 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 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. In my talk, I will give an introduction to the theory of Sigma sets and explain some of these results.