$L^2$-Betti numbers of Dehn fillings

TL;DR

This paper studies the $L^2$-Betti numbers of group-theoretic Dehn fillings, showing they remain unchanged for deep fillings in a broad class of virtually special groups, with applications to geometry and group theory.

Contribution

It introduces the study of $L^2$-Betti numbers in the context of Dehn fillings and proves their invariance in deep fillings for virtually special groups, connecting to geometric conjectures and subgroup structures.

Findings

$L^2$-Betti numbers are preserved in deep Dehn fillings of certain groups.

Verified the Singer Conjecture for specific Einstein manifolds.

Constructed new hyperbolic groups with exotic subgroups from 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.