Quotient and blow-up of automorphisms of graphs of groups

Abstract

In this paper we study the quotient and "blow-up" of graph-of-groups $\mathcal{G}$ and of their automorphisms $H:\mathcal{G}\to \mathcal{G}$. We show that the existence of such a "blow-up" of $\bar{H}:\bar{\mathcal{G}}\to \bar{\mathcal{G}}$ relative to a given family of "local" graph-of-groups isomorphisms $H_i:{\mathcal{G}}_i\to {\mathcal{G}}_i$ depends crucially on the $H_i$-conjugacy class of the correction term $\delta ({\bar{E}}_i)$ for any edge ${\bar{E}}_i$ of $\bar{\mathcal{G}}$, where $H$-congjugacy is a new but natural concept introduced here. As an application we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists to give an actual Dehn twist.

References in corpus (2)

• A Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups
• Hyperbolicity of the Cyclic Splitting Complex