Let \(G\) be a group acting by isometries on a proper geodesic metric space \(X\). Here \(X\) being proper means that every closed bounded subset of \(X\) is compact. Suppose this action is proper and cocompact,. Here, the action is said to be proper if for all compact subsets \(B \subset X\), the set \[\{g \in G | g(B) \cap B \neq \emptyset \}\] is finite. The quotient space \(X/G\) is obtained from \(X\) by identifying any two points \(x, y\) if and only if there exists \(g \in G\) such that \(gx = y\), and equipped with the quotient topology. Then the action of \(G\) on \(X\) is said to be cocompact if \(X/G\) is compact. Under these assumptions, show that \(G\) is finitely generated.

The best solution was submitted by 이정환 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-08.

Alternative solutions were submitted by 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2).