[Paper Review] Algorithmic Problems in Amalgams of Finite Groups

L. Markus-Epstein|ArXiv.org|May 5, 2007

Geometric and Algebraic Topology40 references3 citations

TL;DR

This paper presents an algorithmic framework for solving fundamental decision problems in amalgams of finite groups using generalized Stallings' folding techniques. By constructing subgroup graphs via an extended folding algorithm, the authors achieve polynomial-time solutions for problems including membership, freeness, finite index, and separability, extending methods from free groups to more complex group amalgam structures with proven efficiency and correctness.

ABSTRACT

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.

Motivation & Objective

• To extend Stallings' folding algorithm from free groups to amalgams of finite groups, enabling algorithmic solutions to decision problems.
• To address the challenge of undecidability in general group classes by restricting to amalgams of finite groups, which are hyperbolic and thus admit solvable word problems.
• To develop a graph-theoretic representation of subgroups via labeled graphs (subgroup graphs) that encode subgroup structure and enable efficient computation.
• To provide polynomial-time algorithms for key algorithmic problems such as subgroup presentation, freeness, finite index, and separability in this class of groups.
• To unify and generalize existing methods from free group theory to free products with amalgamation, using inverse automata and graph covers as computational tools.

Proposed method

• Construct a finite, labeled subgroup graph Γ(H) from a finite generating set of a subgroup H in an amalgam G = G₁ ∗ₐ G₂ using a generalized Stallings folding algorithm.
• Apply iterative edge folding and hair-cutting operations to reduce the initial graph of loops into a minimal, reduced precover graph.
• Embed the graph into the Cayley graphs of the free factors G₁ and G₂ by gluing copies of their relative Cayley graphs to components matching the group structure.
• Enforce consistency across the graph by identifying vertices that represent the same group element under the amalgam relation, particularly when paths label elements in A.
• Reduce redundant components (e.g., full Cayley graphs of subgroups of A) to ensure the final graph is minimal and captures only essential subgroup structure.
• Use the resulting graph to read off subgroup properties: loops at the basepoint correspond to elements of H, and structural features of the graph determine subgroup properties like freeness or finite index.

Experimental results

Research questions

• RQ1 Can the Stallings folding method be generalized to solve algorithmic problems in amalgams of finite groups beyond the word problem?
• RQ2 How can subgroup presentations be algorithmically computed using graph-theoretic representations in this class of groups?
• RQ3 What conditions on a subgroup graph allow for efficient detection of freeness, finite index, or separability?
• RQ4 Can the generalized folding algorithm be implemented in polynomial time to solve multiple decision problems simultaneously?
• RQ5 How can the structure of subgroup graphs be used to determine whether a subgroup is malnormal, normal, or trivial?

Key findings

• The generalized Stallings folding algorithm constructs a finite, reduced subgroup graph Γ(H) in quadratic time, which fully encodes the subgroup H in the amalgam G.
• The membership problem for H is solvable in quadratic time by checking for loops labeled by a word at the basepoint in Γ(H).
• The freeness problem is decidable in polynomial time by analyzing the structure of the subgroup graph: H is free if and only if its graph is a tree with no nontrivial cycles.
• The finite index problem is solvable in polynomial time by verifying whether the subgroup graph covers the entire group graph with finite multiplicity.
• The separability problem is decidable in polynomial time by constructing a finite-sheeted cover that separates the subgroup from a given non-element.
• The algorithm provides a systematic way to read off the Kurosh decomposition of a finitely generated subgroup in a free product of finite groups, as shown in companion work [42].