The word problem in verbal products of groups

Javier Brude; Román Sasyk

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arXiv · 2609.31289

The word problem in verbal products of groups

Javier Brude·Román Sasyk

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Abstract

We prove that nilpotent and metabelian products of groups with decidable word problem have decidable word problem. In contrast, we exhibit families of groups with decidable word problem for which their solvable product have undecidable word problem. In doing so, we exhibit hyperbolic groups with undecidable membership problem in its derived subgroups.

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Javier Brude, Román Sasyk. 2026-09-25. The word problem in verbal products of groups. https://arxiv.org/abs/2609.31289

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