A finitely generated group embeds in Thompson's group V if and only if it is the transition group of a context-free graph, which rules out intermediate-growth groups and the Basilica and Hanoi Towers groups.
Groups with Context-free Co-word Problem and Embeddings into Thompson’s GroupV
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A graph-theoretical characterisation of subgroups of Thompson's group $V$
A finitely generated group embeds in Thompson's group V if and only if it is the transition group of a context-free graph, which rules out intermediate-growth groups and the Basilica and Hanoi Towers groups.