We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism. AMS Classification 20E05; 20F28

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construc...

It is not known whether or not the stable rational cohomology groups H̃∗(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions.) We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabili...

Forester has defined spaces of simplicial tree actions for a finitely generated group, called deformation spaces. Culler and Vogtmann’s Outer space is an example of a deformation space. Using ideas from Skora’s proof of the contractibility of Outer space, we show that under some mild hypotheses deformation spaces are contractible. AMS Classification 20E08; 20F65, 20F28

We prove that the reduced C -algebras of centerless mapping class groups and outer automorphism groups of free groups are simple, as are the irreducible pure subgroups of mapping class groups and the analogous subgroups of outer automorphism groups of free groups. r 2003 Elsevier Inc. All rights reserved. MSC: 20F28; 20F65; 46L55; 57N05

Let W be a right-angled Coxeter group. We characterize the centralizer of the Coxeter element of a finite special subgroup of W. As an application, we give a solution to the generalized word problem for Inn(W ) in Aut(W ). Mathematics Subject Classification: 20F10, 20F28, 20F55

We determine the number ω(G) of orbits on the (finite) group G under the action of Aut(G) for G ∈ {PSL(2, q),SL(2, q),PSL(3, 3),Sz(2)}, covering all of the minimal simple groups as well as all of the simple Zassenhaus groups. This leads to recursive formulae on the one hand, and to the equation ω(Sz(q)) = ω(PSL(2, q)) + 2 on the other. MSC 20E32, 20F28, 20G40, 20-04

We examine the palindromic automorphism group ΠA(Fn) of a free group Fn, a group first defined by Collins in [5] which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of SLn(Z), and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which ΠA(Fn) acts properly with finite quo...

A graph G is singular of nullity (> 0), if its adjacency matrix A is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, x, with no zero entries, is called a core graph.We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the en...

The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k . We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomolog...