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15.14 (2002)

Partially Solved

Do there exist finitely generated branch groups (see 15.12)
$\qquad$ a) that are non-amenable?
$\qquad$ b) that are non-amenable and do not contain the free group $F_2$ on two generators?
$\qquad$ c) that contain $F_2$?
$\qquad$ d) that have exponential growth?

Contributor: L. Bartholdi, R. I. Grigorchuk, Z. Šuniḱ

Progress

a), c), d) Such groups do exist (S. Sidki, J. S. Wilson, Arch. Math., 80, no. 5 (2003), 458–463).
b) Yes, there exist finitely generated non-amenable torsion branch groups (S. Kionke, E. Schesler, Bull. London Math. Soc., 56, no. 2 (2024), 536–550).

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