18.39 (2014)

Conjecture: Let $G$ be a hyperbolic group. Then every 2-dimensional rational homology class is virtually represented by a sum of closed surface subgroups, that is, for any $\alpha \in H_2(G; \mathbb{Q})$ there are finitely many closed oriented surfaces $S_i$ and injective homomorphisms $\rho_i : \pi_1(S_i) \to G$ such that $\sum_i [S_i] = n\alpha$, where $[S_i]$ denotes the image of the fundamental class of $S_i$ in $H_2(G)$.