Rational homological stability for groups of symmetric automorphisms of free groups

TL;DR

This paper proves that the rational homology groups of symmetric automorphism groups of free groups stabilize as the rank increases, establishing a form of homological stability.

Contribution

It establishes rational homological stability for symmetric automorphism groups of free groups, a new stability result in the context of automorphism groups.

Findings

Rational homology groups stabilize for large n

Inclusion induces isomorphism in rational homology for n > (3i-1)/2

Provides a new stability result for symmetric automorphisms

Abstract

Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}. An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \Sigma Aut(F_n) be the group of symmetric automorphisms. We prove that the inclusion \Sigma Aut(F_n) \rightarrow \Sigma Aut(F_{n+1}) induces an isomorphism in rational homology for n>(3i-1)/2.