Algorithmic Problems in Amalgams of Finite Groups

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings' methods can be effectively generalized for the class of amalgams of finite groups. In the present paper we employ subgroup graphs constructed by the generalized Stallings' folding algorithm to solve various algorithmic problems in amalgams of finite groups.Comment: 39 pages with 9 figure

Stallings' Foldings and Subgroups of Amalgams of Finite Groups

In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sewing'' on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.Comment: 43 pages with 14 figure

Algorithmic Problems in Amalgams of Finite Groups: Conjugacy and Intersection Properties

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the generalized Stallings' methods, developed by the author, to solve various algorithmic problems concerning finitely generated subgroups of amalgams of finite groups.Comment: 54 pages with 13 figure