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.