Algebra/Topology Seminar

Speaker: Zachary Munro

Title: A lower bound cubical dimension of (Gromov) random groups

Abstract: Gromov defined for a fixed relator length L and density d \in (0,1) a random group presentation. One says that a random group at density d has some property P if the probability that P is true approaches 1 as L approaches infinity. Notably, Ollivier completed a proof of Gromov that for any L and density d < 1/2 a random group is infinite, torsion-free, hyperbolic, with cohomological dimension 2. At density d > 1/2 a random group is either trivial or Z/2Z. For the purposes of this talk, we concern ourselves with random group actions on CAT(0) cube complexes without a global fixed-point. Ashcroft recently proved that random groups at density d < 1/4 act without global fixed-point on a finite-dimensional CAT(0) cube complex. This is conjectured to be an optimal bound. We initiate the investigation into the cubical dimension of random groups by proving that for almost all random groups, any action on a CAT(0) square complex must have a global fixed-point.

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