Summary: THE EULER CHARACTERISTIC OF THE WHITEHEAD
AUTOMORPHISM GROUP OF A FREE PRODUCT
CRAIG JENSEN 1 , JON MCCAMMOND 2 , AND JOHN MEIER 3
Abstract. A combinatorial summation identity over the lattice of labelled hy
pertrees is established that allows one to gain concrete information on the Euler
characteristics of various automorphism groups of free products of groups. In
particular, we establish formulae for the Euler characteristics of: the group
of Whitehead automorphisms Wh(# n
i=1 G i ) when the G i are of finite homo
logical type; Aut(# n
i=1 G i ) and Out(# n
i=1 G i ) when the G i are finite; and the
palindromic automorphism groups of finite rank free groups.
Let G = G 1 # · · · # G n , where the G i are nontrivial groups. There are various
subgroups of Aut(G) and Out(G) (such as the Whitehead automorphism group)
which are only defined with respect to a specified free product decomposition of
G. In this article we calculate the Euler characteristics of several such groups.
Postponing definitions for the moment, our main result is the following.
Theorem A. If G = G 1 # · · · #G n is a free product of groups where # (G) is defined,