### University of Arkansas Topology Seminar: 2/16/2017

### Speaker: Ignat Soroko

### Title: Uncountably many quasi-isometry classes of groups of type FP

An interplay between algebra and topology goes in many ways. Given a space *X*, we can study its homology and homotopy groups. In the other direction, given a group *G*, we can form its Eilenberg−Maclane space *K*(*G*,1). It is natural to wish that it is "small" in some sense. If the space *K*(*G*,1) has *n*−skeleton with finitely many cells, then *G* is said to have type F_{n}. Such groups act naturally on the cellular chain complex of the universal cover for *K*(*G*,1), which has finitely generated free modules in all dimensions up to *n*. On the other hand, if the group ring ℤ*G* has a projective resolution (*P*_{i}) of length *n* where each module *P*_{i} is finitely generated, then *G* is said to have type FP_{n}. There have been many intriguing questions on whether classes F_{n} and FP_{n} are different, and some of them are still open. Bestvina and Brady gave first examples of groups of type FP_{2} which are not finitely presentable (i.e. not of type F_{2}). In his recent paper, Ian Leary has produced uncountably many of such groups. Using Bowditch's concept of taut loops in Cayley graphs, we show that Ian Leary's groups actually form uncountably many classes up to quasi-isometry. This is a joint work with Ian Leary and Robert Kropholler.

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