Largeness and equational probability in groups

TL;DR

This paper introduces concepts of k-genericity and k-largeness in groups, analyzing their thresholds for subsets to cover entire Cartesian powers, and connects these with equational solution measures to advance probabilistic group theory.

Contribution

It defines new notions of largeness in groups and links them with equational probability, providing new results and proofs in equational probabilistic group theory.

Findings

Determines thresholds for k-largeness in group subsets.

Links k-largeness with the measure of solutions to equations.

Provides new insights into equational probabilistic group theory.

Abstract

We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G^n is already the whole of G^n , for various equationally defined subsets. We link this with the inner measure of the set of solutions of an equation in a group, leading to new results and/or proofs in equational probabilistic group theory.