$\mathrm{vb}_2(G)=\sup \{ \dim H_2(H; \mathbb{Q}) \vert H\le G \text {finitely exponential}\} \in \mathbb{Z}_ {\ge 0} \cup \{\infty \}$. _2(G)$ is finite if $ G$ is the fundamental group of closed surfaces and $\text{vb}_2(G)=1$. Extending this result to the limit group and proving that the virtual second Betty number of the limit group $L$ is finite is the fact that $L$ is the fundamental group of free, free abelian, or closed connected surfaces. only if either As an application, we give an alternative proof of Wilton’s result that the surface group is determined between the limit group (and the hyperbolic fundamental group of graphs of free groups with cyclic edge groups) by their profinit complements. An appendix to this paper by Morales explores the relationship between hypothetical his second Betty number and the solubilizing group, and the above results suggest that the group of solubilizing genera in the free and surface groups is the residual explains why he suggests the Jaikin-Zapirain method of showing -$p$ cannot be used for other hyperbolic limit groups.
