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.

# Past Topology Seminar

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.

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).

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.

In directed algebraic topology, a topological space is endowed

with an extra structure, a selected subset of the paths called the

directed paths or the d-structure. The subset has to contain the

constant paths, be closed under concatenation and non-decreasing

reparametrization. A space with a d-structure is a d-space.

If the space has a partial order, the paths increasing wrt. that order

form a d-structure, but the circle with counter clockwise paths as the

d-structure is a prominent example without an underlying partial order.

Dipaths are dihomotopic if there is a one-parameter family of directed

paths connecting them. Since in general dipaths do not have inverses,

instead of fundamental groups (or groupoids), there is a fundamental

category. So already at this stage, the algebra is less desirable than

for topological spaces.

We will give examples of what is currently known in the area, the kind

of methods used and the problems and questions which need answering - in

particular with applications in computer science in mind.

If k is a field of characteristic zero, a theorem of Lurie and Pridham establishes an equivalence between formal moduli problems and differential graded Lie algebras over k. We generalise this equivalence in two different ways to arbitrary ground fields by using “partition Lie algebras”. These mysterious new gadgets are intimately related to the genuine equivariant topology of the partition complex, which allows us to access the operations acting on their homotopy groups (relying on earlier work of Dyer-Lashof, Priddy, Goerss, and Arone-B.). This is joint work with Mathew.

In an article published in 2009, Dave Benson described, for a finite group $G$, the mod $p$ homology of the space $\Omega(BG^\wedge_p)$ --- the loop space of the $p$-completion of $BG$ --- in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. We showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a plus construction of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ is the homology any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson's theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R=F_p$, so that $p$-completion is a special case of the plus construction.

A proper simply connected one-ended metric space is call semi-stable if any two proper rays are properly homotopic. A finitely presented group is called semi-stable if the universal cover of its presentation 2-complex is semi-stable.

It is conjectured that every finitely presented group is semi-stable. We will examine the known results for the cases where the group in question is relatively hyperbolic or CAT(0).

Topological field theories give rise to a wealth of algebraic structures, extending

the E_n algebra expressing the "topological OPE of local operators". We may interpret these algebraic structures as defining (slightly noncommutative) algebraic varieties and stacks, called moduli stacks of vacua, and relations among them. I will discuss some examples of these structures coming from the geometric Langlands program and their applications. Based on joint work with Andy Neitzke and Sam Gunningham.

Answering a question of Milnor, Grigorchuk constructed in the early eighties the

first examples of groups of intermediate growth, that is, finitely generated

groups with growth strictly between polynomial and exponential.

In joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation

2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,

is a growth function of some group. These are the first examples of groups

of intermediate growth where the asymptotic of the growth function is known.

Among applications of our results is the fact that any group of locally subexponential growth

can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be subgroups in Grigorchuk groups).

In a recent work with Tianyi Zheng, we provide near optimal lower bounds

for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by

a measure with power law decay.