We present a construction which associates to a KdV equation the lamplighter group.

In order to establish this relation we use automata and random walks on ultra discrete limits.

It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy

invariants of closed manifolds.

# Topology Seminar

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Let F be a fixed infinite, vertex-transitive graph. We say a graph G is `r-locally F' if for every vertex v of G, the ball of radius r and centre v in G is isometric to the ball of radius r in F. For each positive integer n, let G_n = G_n(F,r) be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties that the random graph G_n has with high probability --- i.e., how these properties depend upon the fixed graph F.

We show that if F is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer r_0 such that for every integer r at least r_0, with high probability the random graph G_n = G_n(F,r) defined above has largest component of size between n^{c_1} and n^{c_2}, where 0 < c_1 < c_2 < 1 are constants depending upon F alone, and moreover that G_n has at least exp(poly(n)) automorphisms. This contrasts sharply with the random d-regular graph G_n(d) (which corresponds to the case where F is replaced by the infinite d-regular tree).

Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics, including a recent and beautiful `rigidity' result of De La Salle and Tessera.

We obtain somewhat more precise results in the case where F is L^d (the standard Cayley graph of Z^d): for example, we obtain quite precise estimates on the number of n-vertex graphs that are r-locally L^d, for r at least linear in d, using classical results of Bieberbach on crystallographic groups.

Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs.

This is joint work with Itai Benjamini (Weizmann Institute).

Two curves in a closed hyperbolic surface of genus g are of the same type if they differ by a mapping class. Mirzakhani studied the number of curves of given type and of hyperbolic length bounded by L, showing that as L grows, it is asymptotic to a constant times L^{6g-6}. In this talk I will discuss a generalization of this result, allowing for other notions of length. For example, the same asymptotics hold if we put any (singular) Riemannian metric on the surface. The main ingredient in this generalization is to study measures on the space of geodesic currents.

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.