I will address two problems about recognizing surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension 2. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognizing surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.
It is a long-standing question whether a group of type F that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type F and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most 17. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I'll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.
The profinite completion of a free profinite group on infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question, which is equivalent to ask: what abstract embedding problems can a free profinite group solve.
Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress.
Let G be a group and S a generating set. Then the group G naturally acts on the Cayley graph Cay(G,S) by left multiplications. The group G is said to be rigid if there exists an S such that the only automorphisms of Cay(G,S) are the ones coming from the action of G. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.
My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups. A group G has spread k if for every g1,…,gk we can find an h in G such that ⟨gi,h⟩=G. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread 0. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven's first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we'll see the answer to this question. The arguments will be concrete and accessible to a general audience.
In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on n > 1 generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means 'as rich as possible', and furthermore that the free group can be made dense in the ambient full group of the equivalence relation.
Recent work in the theory of locally compact second-countable (l.c.s.c.) groups has highlighted the importance of chief factors, meaning pairs of closed normal subgroups K/L such that no closed normal subgroups lie strictly between K and L. In particular, the group K/L is then topologically characteristically simple, meaning it has no proper nontrivial closed subgroup that is preserved by all automorphisms. I will present a classification of the abelian l.c.s.c. topologically characteristically simple groups: these all occur as chief factors of soluble groups, and naturally fall into five families with a few parameters. Each family has a straightforward characterization within the class of abelian l.c.s.c. groups, without directly invoking the property of being topologically characteristically simple.
Let G be a group acting on a regular tree. The 'local' actions that vertex stabilisers in G induce on balls around the fixed vertex are innately connected to the 'global' structure of G. I demonstrate this relationship and define a particularly accessible class of groups acting on (locally finite) regular trees by 'prescribing' said local actions, following Burger-Mozes. Being defined solely in terms of finite permutation groups, these groups allow us to introduce computational methods to the world of locally compact groups: I will outline the capabilities of a recently developed GAP package that provides methods to create, analyse and find suitable local actions.
We will survey a series of recent developments in the area of first-order descriptions of linear groups. The goal is to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups. We also dwell on the principal problem of isotipicity of finitely generated groups.
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