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In the 1980's Stallings showed that every finitely generated subgroup of afree group is canonically represented by a finite minimal immersion of abouquet of circles. In terms of the theory of automata, this is a minimalfinite inverse automaton. This allows for the deep algorithmic theory of finiteautomata and finite inverse monoids to be used to answer questions aboutfinitely generated subgroups of free groups.  In this paper we attempt to apply the same methods to other classes ofgroups. A fundamental new problem is that the Stallings folding algorithm mustbe modified to allow for ``sewing'' on relations of non-free groups. We look atthe class of groups that are amalgams of finite groups. It is known that thesegroups are locally quasiconvex and thus all finitely generated subgroups arerepresented by finite automata. We present an algorithm to compute such afinite automaton and use it to solve various algorithmic problems.

Stallings' Foldings and Subgroups of Amalgams of Finite Groups