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$L^2$-Betti numbers of Dehn fillings

Nansen Petrosyan, Bin Sun

TL;DR

This work develops a comprehensive framework for analyzing $L^2$-Betti numbers under group-theoretic Dehn fillings in the setting of virtually compact cubical groups hyperbolic relative to virtually abelian peripherals. By constructing a Dehn filling space and leveraging quantitative Lück approximation, Cohen-Lyndon theory, and the Atiyah conjecture, the authors prove that sufficiently deep Dehn fillings preserve $L^2$-Betti numbers, and in broad cases even yield equalities $ ext{b}^{(2)}_*(\overline{G})=\text{b}^{(2)}_*(G)$. The results extend to multiple peripheral subgroups and, via virtually sparse special groups, to cusped arithmetic lattices, enabling applications to Singer conjecture instances for Einstein manifolds, algebraic fibering, and the creation of exotic hyperbolic subgroups. They also provide concrete deficiency bounds for Dehn fillings of non-uniform lattices, connecting topological invariants to group-theoretic structure through the Cohen–Lyndon framework and the Malnormal/Quasi-convex machinery of modern geometric group theory.

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.

$L^2$-Betti numbers of Dehn fillings

TL;DR

This work develops a comprehensive framework for analyzing -Betti numbers under group-theoretic Dehn fillings in the setting of virtually compact cubical groups hyperbolic relative to virtually abelian peripherals. By constructing a Dehn filling space and leveraging quantitative Lück approximation, Cohen-Lyndon theory, and the Atiyah conjecture, the authors prove that sufficiently deep Dehn fillings preserve -Betti numbers, and in broad cases even yield equalities . The results extend to multiple peripheral subgroups and, via virtually sparse special groups, to cusped arithmetic lattices, enabling applications to Singer conjecture instances for Einstein manifolds, algebraic fibering, and the creation of exotic hyperbolic subgroups. They also provide concrete deficiency bounds for Dehn fillings of non-uniform lattices, connecting topological invariants to group-theoretic structure through the Cohen–Lyndon framework and the Malnormal/Quasi-convex machinery of modern geometric group theory.

Abstract

We initiate the study of the -Betti numbers of group-theoretic Dehn fillings. For a broad class of virtually special groups , we prove that the -Betti numbers of sufficiently deep Dehn fillings are equal to those of . As applications, we verify the Singer Conjecture for certain Einstein manifolds, establish a virtual fibering criterion for , obtain bounds on deficiency of , 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.

Paper Structure

This paper contains 23 sections, 62 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finiteness properties
  4. Von Neumann rank of a matrix
  5. $L^2$-Betti numbers and (Strong) Atiyah Conjecture
  6. Hyperbolically embedded subgroups
  7. Dehn filling
  8. Quantifying Lück's approximation
  9. The first $L^2$-Betti number
  10. Dehn filling space and $L^2$-Betti numbers
  11. Dehn filling space
  12. Bounding finite subgroups
  13. Compact special Dehn filling
  14. Main results for multiple peripherals
  15. Dehn filling space
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be a virtually compact cubical group that is hyperbolic relative to a virtually abelian subgroup $H\leqslant G$. Then there exists a finite-index torsion-free normal subgroup $K\lhd H$ such that for every sufficiently deep normal subgroup $N\lhd H$ that satisfies $N\leqslant K$ and $K/N$ is

Figures (1)

  • Figure 1: Dehn filling space associated to the triple $(G,H,N)$. The mappings cylinders $M_{\psi}$ and $M_{\phi}$ are shaded blue and grey, respectively. Their bases $B\overline{H}$ and $BG$ are represented by larger discs. The common subspace $BH$ is shaded red.

Theorems & Definitions (144)

  • Theorem 1.1: \ref{['thm. sparse rel. hyp.']}
  • Theorem 1.2: see also \ref{['cor. gen li']}
  • Theorem 1.3: \ref{['thm. hyp.']} (see also \ref{['thm. gen hyp.']})
  • Remark 1.4
  • Corollary 1.5: \ref{['cor. 1 fiber']}
  • Corollary 1.6: \ref{['cor. k fiber']}
  • Corollary 1.7: \ref{['cor. constructing Einstein']}
  • Theorem 1.8: \ref{['thm. exotic quotient']}
  • Corollary 1.10: \ref{['thm. lattice summarize']} \ref{['item. lattice li']}
  • Corollary 1.11: \ref{['cor. defi lattice']}
  • ...and 134 more