A note on the R∞ property for groups FAlt(X)⩽G⩽Sym(X)

Abstract

Given a set X, the group consists of all bijective maps from X to X, and is the subgroup of maps with finite support i.e. those that move only finitely many points in X. We describe the automorphism structure of groups and use this to state some conditions on G for it to have the property. Our main results are that if G is infinite, torsion, and , then it has the property. Also, if G is infinite and residually finite, then there is a set X such that G acts faithfully on X and, using this action, has the property. Finally we have a result for the Houghton groups, which are a family of groups we denote Hn, where . We show that, given any , any group commensurable to Hn has the property.