Pure Maths Colloquium - QI rigidity of groups containing almost normal subgroups - Dr Alex Margolis (Technion) Seminar

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Abstract: A goal of geometric group theory is to understand to what extent the large scale geometry of a finitely generated group determines its algebraic structure. A subgroup H of G is said to be almost normal if every conjugate of H is commensurable to H. If G is finitely generated and H is almost normal, then G can be thought of as a coarse bundle over the coset space G/H. We show that quasi-isometries frequently preserve almost normal subgroups and the associated coarse bundle structure. A sample application is the following: any finitely presented group quasi-isometric to a Z-by-(∞ ended) group is also Z-by-(∞ ended). We make use of the notion of relative ends due to Kropholler and Roller. Our results build on work of Dunwoody and Mosher-Sageev-Whyte.