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University of Sydney Algebra Seminar

Michael Björklund

Friday 1 March, 12-1pm, Place: Carslaw 175

Quasi-morphisms and approximate lattices

An approximate lattice is a uniformly discrete approximate subgroup $\mathrm{\Lambda }$ of a locally compact group $G$ for which there is a finite volume Borel set $B$ in $G$ such that $B\mathrm{\Lambda }=G$. To every such approximate lattice, one can associate a dynamical system of $G$, which, in the case when $\mathrm{\Lambda }$ is a lattice coincides with the canonical $G$-action on the quotient space $G/\mathrm{\Lambda }$. In this talk we discuss how one can construct approximate lattices from (cohomologically non-trivial) quasi-morphisms, and show that the corresponding (compact) hulls do not admit any invariant probability measures, and always project to a non-trivial Furstenberg boundary. No prior knowledge of approximate lattices or quasi-morphisms will be assumed. Based on joint work with Tobias Hartnick (Karlsruhe).