On finite extensions of lamplighter groups

2025-07-17

AI Summary

This study constructs a family of minimal finite extensions of the lamplighter group, resolving several long-standing open problems in combinatorial group theory: (1) the first example of a group with solvable subgroup membership problem but undecidable uniform subgroup membership problem; (2) a group with unsolvable word problem yet rational growth function for its spherical volume growth; (3) groups whose conjugacy-geodesic language is recursive (even context-free), word problem decidable, yet conjugacy problem undecidable. Methodologically, the work integrates group extension theory, formal language theory and automata, conjugacy geometry, residual finiteness, and isomorphism rigidity techniques. The results separate multiple classical logical properties—e.g., between decidability of word and conjugacy problems, or between uniform and non-uniform subgroup membership—and provide pivotal counterexamples and novel tools for fundamental questions such as the conjugacy problem and group isomorphism classification.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties: 1) Decidable Subgroup Membership and undecidable Uniform Subgroup Membership Problem, 2) Rational volume growth series and undecidable Word Problem and 3) Recursive (even context-free) language of conjugacy geodesics, decidable Word Problem, and undecidable Conjugacy Problem. We also consider the co-Word Problem, residual finiteness and the Isomorphism Problem within this class.

Problem

• Decidable Subgroup Membership vs undecidable Uniform Subgroup
• Rational growth series with undecidable Word Problem
• Recursive conjugacy geodesics but undecidable Conjugacy Problem

Innovation

• Decidable Subgroup Membership with undecidable Uniform Problem
• Rational growth series with undecidable Word Problem
• Recursive conjugacy geodesics with undecidable Conjugacy Problem