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Geometric invariants of discrete and locally compact groups

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.