Rice University Department of Mathematics Colloquium

Curvature, rigidity and finite shadows of infinite groups

4:00 pm Thursday, April 12, 2018
Martin Bridson (Oxford)

Abstract: There are many situations in geometry and group theory where it is natural or convenient to explore infinite groups via their actions on finite objects -- ie via their finite quotients. But how hard is it find finite quotients, and to what extent do they determine the group? In this lecture I'll outline the great advances of recent years in this area. I'll describe pairs of distinct groups that have the same finite quotients and I'll sketch the proof of some "profinite rigidity results", ie theorems showing that in certain circumstances one can identify an infinite group if one knows its set of finite quotients. I'll emphasize how an enhanced understanding of spaces of non-positive curvature has underpinned progress on key algebraic questions, and pay particular attention to 3-dimensional manifolds, where a remarkable mingling of arithmetic and geometry leads to profinite rigidity theorems. I'll outline recent work with Reid, McReynolds and Splitler that yields the first full-sized groups that are profinitely rigid in the absolute sense.

Return to Colloquium page