Rational homological stability for groups of partially symmetric automorphisms of free groups

TL;DR

This paper establishes rational homological stability results for groups of partially symmetric automorphisms of free groups, showing stability under increasing the rank or the number of partially symmetric generators.

Contribution

It proves new rational homological stability theorems for groups of partially symmetric automorphisms of free groups, extending understanding of their algebraic topology.

Findings

Rational homology stabilizes as the rank increases for fixed m.

Rational homology stabilizes as the number of partially symmetric generators increases for fixed n.

Stability holds in specific dimension ranges depending on n and m.

Abstract

Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}. An automorphism \phi of F_{n+m} is called partially symmetric if for each 1 \le i \le m, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let \Sigma\Aut_n^m be the group of partially symmetric automorphisms. We prove that for any m \ge 0 the inclusion \Sigma\Aut_n^m \to \Sigma\Aut_{n+1}^m induces an isomorphism in rational homology for dimensions i satisfying n \ge (3(i+1)+m)/2, with a similar statement for the groups P\Sigma\Aut_n^m of pure partially symmetric automorphisms. We also prove that for any n \ge 0 the inclusion \Sigma\Aut_n^m \to \Sigma\Aut_n^{m+1} induces an isomorphism in rational homology for dimensions i satisfying m > (3i-1)/2.