Aspects of the coarse geometry of discrete groups

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The proposed research program aims to advance the understanding of discrete groups by regarding them as geometric objects. This is part of Gromov's program of studying quasi-isometry properties of groups and their relations to algebraic properties. The proposal has particular emphasis on groups acting on non-positively curved spaces as hyperbolic and relatively hyperbolic groups, small cancellation groups, CAT(0) groups, among others. The proposal has three general objectives:***1. To understand the classes of nonpositively curved groups that are closed under taking finitely presented subgroups and the relation to dimension. In particular, to identify combinatorial conditions on G-complexes implying properties of the subgroup structure of the group G. We propose to use techniques from algebraic topology that have not been fully explored in the study of coherence and local quasiconvexity. Specifically, the use of L^2 Betti numbers in the study of coherence and local quasiconvexity, and the use of Bredon modules over the orbit category in relation with homological isoperimetric inequalities. These are tools that the Principal Investigator (and its collaborators) have recently introduced to the study of subgroups of non-positively curved groups. The use of these tools is novel in the area and several aspects remain to be explored. We expect our investigations to shed some light into outstanding questions in the area as residual finiteness.***2. To advance the study of homological approaches to define quasi-isometry invariants. The emphasis here is to continue to develop the theory of homological higher dimensional Dehn functions. Recently, Hanlon and the PI exhibited an algebraic approach to these invariants and used it to obtain results on the subgroup structure of certain classes of discrete groups. There are several directions to further develop the study of these invariants in connection with the study of subgroups of discrete groups. We expect that our algebraic approach to filling functions will reveal new connections between homological and coarse geometric group invariants.***3. Classical combinatorial games on graphs have versions that yield quasi-isometry invariants of infinite graphs, and hence invariants of finitely generated groups (via Cayley graphs). The relation between these quasi-isometry invariants and the theory of discrete groups is mostly unexplored. We plan to investigate these relations. Current work in progress suggests new characterizations of hyperbolic groups; relations between splittings of groups and containment games; and certain aspects of amenability seem to be related to particular games. These investigations will create bridges between the community in game theory on graphs, and geometric group theorists.