2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62916We examine the palindromic automorphism group $ΠA(F_n)$ 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 $ΠA(F_n)$ acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of $ΠA(F_n)$, such as that its rational cohomology is zero at the vcd.Group TheoryAlgebraic Topology20F32, 20J05; 20F28, 55N91text