Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on complex hyperbolic lattices and strict hyperbolization

Kejia Zhu

Abstract

We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney--Davis and Davis--Januszkiewicz--Weinberger. Viewing a non-uniform lattice $Γ$ in $\text{PU}(n,1)$ as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that when $n\geq 2$, $Γ$ is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont--Ruffoni. We also prove that a uniform lattice in $\text{PU}(n,1)$ is not the fundamental group of a Charney-Davis strict hyperbolization when $n\geq 2$, assuming the initial complex satisfies some mild conditions.

A note on complex hyperbolic lattices and strict hyperbolization

Abstract

We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney--Davis and Davis--Januszkiewicz--Weinberger. Viewing a non-uniform lattice in as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that when , is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont--Ruffoni. We also prove that a uniform lattice in is not the fundamental group of a Charney-Davis strict hyperbolization when , assuming the initial complex satisfies some mild conditions.
Paper Structure (8 sections, 8 theorems, 17 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relatively Hyperbolic Dehn Filling
  3. Compactification of $M$
  4. The structure of cusps
  5. Borel-Serre compactification
  6. Toroidal compactification
  7. Collapsing the cusps
  8. Proof of the Theorem \ref{['maintheoremnew']}

Key Result

Theorem 1.1

The process of strict hyperbolization of a compact homogeneous $n$-dimensional simplicial complex without boundary and the process of relative strict hyperbolization of a finite simplicial complex with subcomplex cannot produce complexes with fundamental groups isomorphic to complex hyperbolic latti

Figures (1)

  • Figure 1: Schematic picture of the 3 compactifications of $M$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.4: groves2023relative-Theorem A
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof
  • Remark 1.8
  • Definition 2.1: Dehn Filling, GrovesManningDehn
  • Definition 2.2
  • ...and 9 more