Semi-Direct Products of Graphs of Groups
by
Debra Boutin
Hamilton College
Coauthors: Thomas A Stiadle (Wells College)

Mathematicians have long used automorphisms of a graph to learn about automorphisms of its fundamental group. The Realization Theorem tells us that every finite subgroup of Aut(F_n) shows up as a group of automorphisms of a finite graph whose fundamental group is F_n and thus characterizes the subgroups of Aut(F_n) that can be realized by automorphisms of a graph. This talk introduces work that generalizes this idea to graphs of groups. To learn more about automorphisms of a graph of groups and what they tell us about the automorphisms of the fundamental group we define an action of one graph of groups H on another G, and a semi-direct product G\rtimes H (which is itself a graph of groups). This talk will show how the groups \pi₁(G) and \pi₁(G\rtimes H) are related and the conditions under which \pi₁(G\rtimes H) \slash \pi₁(G) makes sense as a subgroup of Aut(G) and Aut(\pi₁(G)).

Date received: September 19, 2000