Largeness and equational probability in groups

Alexander Varieties and Largeness of Finitely Presented Groups

Let X be a finite CW complex. We show that the fundamental group of X is large if and only if there is a finite cover Y of X and a sequence of finite abelian covers {YN} of Y which satisfy b1(YN ) ≥ N . We give some applications of this result to the study of hyperbolic 3–manifolds, mapping classes of surfaces and combinatorial group theory.

Largeness and Sq-Universality of cyclically Presented Groups

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contain, amongst others, the various generalizations of Fibonacci groups introduce...

Simple Groups, Permutation Groups, and Probability

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not entirely anticipated by the nature of th...

Mixing and Triple Recurrence in Probability Groups

We consider a class of groups equipped with an invariant probability measure (call them probability groups), which includes all compact groups and is closed under taking ultraproducts with the induced Loeb measure. This note develops the basics of the theory of measure-preserving actions of these groups on probability spaces, culminating in a triple recurrence result for mixing probability grou...

Probability on Compact Lie Groups