Solution sets of equations in infinite discrete groups

Abstract

Solution sets of equations in groups can be thought of as group-theoretic analogues of algebraic varieties. In this thesis we apply methods of geometric group theory to study such solution sets in infinite groups from probabilistic and algebraic points of view. This is a 'three paper thesis', the main body of which consists of the following papers:
[1] M. Valiunas, Rational growth and degree of commutativity of graph products, J. Algebra 522 (2019), 309{331.
[2] A. Martino, M. Tointon, M. Valiunas, and E. Ventura, Probabilistic nilpotence in infinite groups, preprint, available at arXiv:1805.11520 [math.GR], 2018.
[3] M. Valiunas, Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products, preprint, available at arXiv:1811.02975 [math.GR], 2018.

In [1], we study graph products of groups - a generalisation of direct and free products. We use regularity of growth of certain graph products to establish bounds on sizes of spheres in Cayley graphs of such groups G. We use these bounds to show that in a graph product that is not virtually abelian, two randomly chosen elements inside a large ball in a Cayley graph of G will 'almost never' commute - that is, the solution set of the equation [X1;X2] = 1 is negligible.
In [2], we use similar methods to study higher commutators, that is, the equations [X1; : : : ;Xk+1] = 1. In particular, we show that for most classes of groups that are not virtually k-step nilpotent, the (k + 1)-fold simple commutator of randomly chosen elements will almost never be trivial. Here, 'randomly chosen' refers either to using sequences of measures on a finitely generated group that are well-behaved with respect to finite-index subgroups, or to looking at finite quotients of a residually finite group. We also analyse regularity of the solution set of such an equation in virtually k-step nilpotent groups, and produce examples of groups showing necessity of our assumptions on finite generation or residual finiteness.
In [3], we study negative curvature in graph products of groups. We use quasi-median graphs - a class of 'nonpositively curved' graphs generalising the notion of CAT(0) cube iii iv complexes - to construct explicit acylindrical actions of graph products on spaces quasi-isometric to trees. We use this action to show that, given a finite collection of groups Gi with the property that any system of equations in Gi is equivalent to a finite subsystem, certain graph products of the Gi will also satisfy this property.

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Identifiers

ORCID for Motiejus Valiunas: orcid.org/0000-0003-1519-6643

ORCID for Armando Martino: orcid.org/0000-0002-5350-3029

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