Logic Seminar
Christian Rosendal
UIC
Large scale geometry of metrisable groups
Abstract: Large scale geometry on the one hand originates in Banach space theory, notably by work of Enflo and Ribe, while in the setting of countable discrete groups is mainly an invention due to Gromov. The fundamental observation here is that the word metric on a finitely generated group is left-invariant and, while not unique since it involves a choice of finite generating set, it is independent of the generating set up to quasi-isometry. In the setting of locally compact second countable groups the situation is essentially equivalent since, by a result of Struble, every compactly generated G group admits a compatible left-invariant proper metric quasi-isometric to the word metric of a compact generating set.
However, if G is a metrisable topological group, though G admits many left-invariant metrics, it is not clear that there should be any manner of defining a unique quasi-isometry type since there is no canonical generating set for G. Nevertheless, we show how to overcome this problem by considering a notion of "metric compactness", which we enable us to compute the intrinsic quasi-isometric type of various such groups. We also present associated results for metrically proper affine actions on Banach spaces and discuss how our theory plays out for automorphism groups of countable first-order structures.
Tuesday January 21, 2014 at 4:00 PM in SEO 427