Are there any sufficient conditions in geometric group theory for a group to be finite? Are there any necessary conditions?
2026-03-26 13:00:46.1774530046
Conditions for finiteness of group in geometric group theory
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Often in Geometric Group Theory, we study a group by studying a space it acts on in a certain way. A typical way we like groups to act on spaces is via a geometric group action, that is, an action that is:
We typically want this action to be on a proper, geodesic metric space. Such an action has a very simple description of when a group is finite: If a finitely-generated group $G$ acts geometrically on a proper, geodesic metric space, then the group is finite if and only if the metric space is of finite diameter.