Homology of free loop space

Question

By rational homotopy theory, $H(\Lambda M; \mathbb{Q})$ is infinite-dimensional over $\mathbb{Q}$ if $M$ is simply-connected. Are there (non-simply-connected) examples when $H(\Lambda M; \mathbb{Q})$ is finite-dimensional? I am most interested when $M$ is a manifold. Also, note that $H_0(\Lambda M; \mathbb{Q})$ is the set of conjugacy classes of $\pi_1(M)$, ie. loops up to homotopy.

Answer 1

Take a nontrivial (say, infinite) group $G$ which has exactly two conjugacy classes (e.g. one given by the HNN construction). Now, take $X=K(G,1)$. By working harder one can construct examples of such groups which have finite cohomological dimension and, hence, are fundamental groups of (noncompact) aspherical manifolds.

Edit. In fact, if you apply the HNN construction to the infinite cyclic group $G_0$, the result will an (infinitely generated) countable group $G$ of cohomological dimension 2 with exactly two conjugacy classes (the presentation complex $X$ of $G$ will be 2-dimensional and aspherical). Hence, by Stallings embedding theorem, there exists a 4-dimensional aspherical manifold $M=K(G,1)$ (obtained by embedding $X'$ homotopy-equivalent to $X$ into $R^4$ and then taking a regular neighborhood there).

Lastly, there are no known examples of infinite finitely presented groups with finitely many conjugacy classes.