A single family of finite extensions of lamplighter groups separates uniform from non-uniform subgroup membership, pairs rational growth with an undecidable word problem, and pairs a context-free conjugacy geodesic language with an undecidable conjugacy problem.
Finite generating sets of relatively hy- perbolic groups and applications to geodesic languages
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On finite extensions of lamplighter groups
A single family of finite extensions of lamplighter groups separates uniform from non-uniform subgroup membership, pairs rational growth with an undecidable word problem, and pairs a context-free conjugacy geodesic language with an undecidable conjugacy problem.