2020-08 Geometric action revisited

In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.

Would this conclusion still hold in the case the space X is not necessarily proper?

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About Hyungryul

2003.3-2009.8 KAIST, Undergraduate student in Mathematics 2009.8-2014.8 Cornell University, PhD student in Mathematics 2014.9-2017.2 University of Bonn, Postdoc 2019.3-Present KAIST, Assistant Professor