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15.42 (2002)

Open

Is it true that the group algebra $k[F]$ of R. Thompson’s group $F$ (see 12.20) over a field $k$ satisfies the Ore condition, that is, for any $a, b \in k[F]$ there exist $u, v \in k[F]$ such that $au = bv$ and either $u$ or $v$ is nonzero? If the answer is negative, then $F$ is not amenable.

Contributor: V. S. Guba

Progress

Comment of 2026: If the answer is positive, then $F$ is amenable (so problem 15.42 is equivalent to 12.20), because the “if and only if” statement holds for groups satisfying Kaplansky’s zero-divisor conjecture (D. Kielak, Preprint, 2026, https://arxiv.org/pdf/1605.09133v2), and $F$ does satisfy it.

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