In my talk, I will explain the main ideas of how to prove that one-relator groups and their group algebras over fields of characteristic zero are coherent.
Tag - One-relator groups
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.
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.
One-relator groups G=F/≪w≫ pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints (eg Magnus's theorem that the word problem is soluble). On the other hand, they are not susceptible to geometric techniques, since some of them (such as Baumslag-Solitar groups) exhibit extremely pathological behaviour.
I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder - the primitivity rank π(w), the smallest rank of a subgroup of F containing w as an imprimitive element. A sample application is that every subgroup of G of rank less than π(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.
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