Mixed identities, hereditarily separated actions and oscillation

TL;DR

This paper establishes general conditions under which equations with parameters over certain topological G-spaces have solutions, with applications to actions of Thompson's group F and branch groups, advancing understanding of equations in group actions.

Contribution

It introduces broad criteria for solvability of parameterized equations in topological G-spaces, with specific applications to Thompson's group F and branch groups.

Findings

Conditions guaranteeing solutions to equations with parameters

Applications to Thompson's group F actions

Applications to branch group actions

Abstract

Given a topological $G$-space we consider equations with parameters over $G$. In particular we formulate some very general conditions on words with parameters $w(\bar{y},\bar{g})$ over $G$ which guarantee that the inequality $w(\bar{y},\bar{g})\neq 1$ has a solution in $G$. These results are illustrated in some typical situations, in particular standard actions of Thompson's group $F$ and branch groups are considered. The major results of this paper appeared in some form in Section 2 of the PhD thesis of the second author (avalable at arXiv:1308.6330).