 
Summary: THE PURE SYMMETRIC AUTOMORPHISMS OF
A FREE GROUP FORM A DUALITY GROUP
NOEL BRADY, JON MCCAMMOND, JOHN MEIER, AND ANDY MILLER
Abstract. The pure symmetric automorphism group of a finitely generated
free group consists of those automorphisms which send each standard generator
to a conjugate of itself. We prove that these groups are duality groups.
1. Introduction
Let Fn be a finite rank free group with fixed free basis X = {x1, . . . , xn}. The
symmetric automorphism group of Fn, hereafter denoted n, consists of those au
tomorphisms that send each xi X to a conjugate of some xj X. The pure
symmetric automorphism group, denoted Pn, is the index n! subgroup of n of
symmetric automorphisms that send each xi X to a conjugate of itself. The
quotient of Pn by the inner automorphisms of Fn will be denoted OPn. In this
note we prove:
Theorem 1.1. The group OPn is a duality group of dimension n  2.
Corollary 1.2. The group Pn is a duality group of dimension n  1, hence n
is a virtual duality group of dimension n  1.
(In fact we establish slightly more: the dualizing module in both cases is
free.)
