$L^2$-Betti numbers of Dehn fillings

Abstract

We initiate the study of the $L^2$-Betti numbers of group-theoretic Dehn fillings. For a broad class of virtually special groups $G$, we prove that the $L^2$-Betti numbers of sufficiently deep Dehn fillings $\overline{G}$ are equal to those of $G$. As applications, we verify the Singer Conjecture for certain Einstein manifolds, establish a virtual fibering criterion for $\overline{G}$, obtain bounds on deficiency of $\overline{G}$, and provide new examples of hyperbolic groups with exotic subgroups that arise as Dehn fillings of any cusped arithmetic hyperbolic manifold of dimension at least four.

54 pages, 1 figure. Theorem 1.8 has been strengthened to show that every cusped arithmetic hyperbolic manifold of dimension at least 4 gives rise to an infinite family of hyperbolic groups with exotic subgroups in each case. There are also minor changes and rearrangements in the introduction