In this talk I will discuss recent joint work with Dominik Gruber where

we find a reasonable model for random (infinite) Burnside groups,

building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the

quotient of a free group of rank r by the normal subgroup generated by

all elements of the form g^n; quotients of such groups are called

Burnside groups. In 1902, Burnside asked whether any such groups could

be infinite, but it wasn't until the 1960s that Novikov and Adian showed

that indeed this was the case for all large enough odd n, with later

important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic

groups in the 1980s and 90s, he observed that random quotients of free

groups have interesting properties: depending on exactly how one chooses

the number and length of relations one can typically gets hyperbolic

groups, and these groups are infinite as long as not too many relations

are chosen, and exhibit other interesting behaviour. But one could

equally well consider what happens if one takes random quotients of

other free objects, such as free Burnside groups, and that is what we

will discuss.

# Topology Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

In this talk I will articulate and contextualize the following sequence of results.

The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group.

Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories.

In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories.

This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis.

I will introduce two obstructions for a rational homology 3-sphere to smoothly bound a rational homology 4-ball- one coming from Donaldson's theorem on intersection forms of definite 4-manifolds, and the other coming from correction terms in Heegaard Floer homology. If L is a nonunimodular definite lattice, then using a theorem of Elkies we will show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction coming from Donaldson's theorem. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.