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21.105 (2026)

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(D. Segal). A group word $w$ is said to be concise in a class $\mathcal{C}$ of groups if for every group $G$ in $\mathcal{C}$ such that the set $G_w$ of word values of $w$ in $G$ is finite, the verbal subgroup $w(G) = \langle G_w \rangle$ is also finite (see also 2.45). Is every word concise in the class of residually finite groups?

Contributor: M. Petschick

Progress

See (D. Segal, Words. Notes on verbal width in groups, Cambridge Univ. Press, 2009) for a partial solution.

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