### The space of marked groups and markings of finite groups

Analysis and Geometry Seminar

22nd May 2018, 3:00 pm – 4:00 pm

Howard House, 2nd Floor Seminar Room

This talk is partly based on joint work with Hiroki Sako (Niigata University). For each fixed natural number k, Grigorchuk studied the space of all pairs of k-generated groups and k-generating sets (each such pair is called a k-marked group). It is naturally endowed with a compact metrizable topology. It has been discovered by several researchers that for an infinite sequence of k-generated finite groups, some property of the sequence with a specific system of k-markings (for instance, the property of being expanders, or a coarse geometric property of the disjoint union of Cayley graphs) is seriously affected by the choice of systems. We study such dependence in connection with accumulation points in the space of k-marked groups.

As an application, we construct two dense finitely generated subgroups in a certain fixed compact group with the following features: One is constructed by adding one extra generator to a generating set of the other; yet group properties of them are considerably different.

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