Partial Word and Equality problems and Banach densities

Abstract

We investigate partial Equality and Word Problems for finitely generated groups. After introducing Upper Banach (UB) density on free groups, we prove that solvability of the Equality Problem on squares of UB-generic sets implies solvability of the whole Word Problem. In particular, we prove that solvability of generic EP implies WP. We then exploit another definition of generic EP, which turns out to be equivalent to generic WP. We characterize in different ways the class of groups with unsolvable UB-generic WP, proving that it contains that of algorithmically finite groups, and it is contained in that of groups with unsolvable generic WP.