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15.8 (2002)

Partially Solved

(S. M. Ulam). Let $G$ be the compact group $SO(3)$ of all rotations of a 3-dimensional Euclidean space, viewed as a discrete group.
$\qquad$ a) Can $G$ act non-trivially on a countable set?
$\qquad$ b) Let $G$ be any Lie group (indeed, any separable continuous group) made discrete: can $G$ act faithfully on a countable set?

Contributor: P. de la Harpe

Progress

a) Yes, it can (S. Thomas, J. Group Theory, 2 (1999), 401–434); another proof is in (Yu. L. Ershov, V. A. Churkin, Dokl. Math., 70, no. 3 (2004), 896–898).

b) Comment of 2021: an affirmative answer was obtained for the nilpotent case (N. Monod, J. Group Theory, 25, no. 5 (2022), 851–865). Comment of 2025: this problem is reduced to the case of simple Lie groups; in particular, the solvable case is settled (A. Conversano, N. Monod, J. Algebra, 640 (2024), 106–116).

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