Talk:L2-Betti numbers for the universal covering of the circle (Ex)
I don't feel comfortable writing the solution here since I have seen before. But if you are stuck, here are possible suggestions, each an independent way to solve it:
0. You could just calculate the homology, it's very easy.
1. has nice behavior from a functional analytic perspective, and . You can determine its group von Neumann algebra very explicitly. See Luck's book L2-invariants, Example 1.4.
2. The fundamental group of is very nice and Slide 4 of Luck's slides has an applicable theorem. (Also Lemma 1.34 from his book.)
3. homology satisfies various homotopy-theoretic properties; see Theorem 1.35 in Luck's book. Then a solution that is both very stupid and very amusing is to use Lott-Luck, in Slide 5, with a 3-manifold with appropriate boundary.