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7.36 (1980)

Solved

Is it true that every residually finite group in which every subgroup of finite index (including the group itself) is defined by a single defining relation is either free or isomorphic to the fundamental group of a compact surface?

Contributor: O. V. Mel’nikov

Progress

No; for example, let $H_n = \langle x, y \mid y^{-1}xy = x^n \rangle$, $n = 2, 3, \dots$; then every subgroup of finite index in $H_n$ is isomorphic to a group $H_m$ for some $m$ (V. A. Churkin, Abstracts of 8th All-USSR Symp. on Group Theory, Kiev, 1982, 139–140 (Russian)).

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