Geometry for palindromic automorphism groups of free groups

Abstract

We examine the palindromic automorphism group $\Pi A(F_n)$ of a free group $F_n$, a group first defined by Collins which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of $SL_n(\Z)$, and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which $\Pi A(F_n)$ acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of $\Pi A(F_n)$, such as that its rational cohomology is zero at the vcd.