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A note on complex hyperbolic lattices and strict hyperbolization

  • Kejia Zhu EMAIL logo
Published/Copyright: May 23, 2025
Journal of Group Theory
From the journal Journal of Group Theory Volume 28 Issue 6

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 PU ⁢ ( n , 1 ) as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that, when n ≥ 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 PU ⁢ ( n , 1 ) is not the fundamental group of a Charney–Davis strict hyperbolization when n ≥ 2 , assuming the initial complex satisfies some mild conditions.

Acknowledgements

The author would like to thank his advisor, Daniel Groves, for answering his questions. He also would like to thank Corey Bregman, Matthew Durham, Shaver Phagan, and Lorenzo Ruffoni for their helpful comments. In particular, the author would like to thank Corey Bregman for permitting the author to employ some methods and techniques from our joint work in an unpublished version of [4] which are not used in the final version with Daniel Groves. The author is also grateful to the referee for many helpful comments that improved the paper.

  1. Communicated by: Adrian Iona

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Received: 2024-09-24
Revised: 2025-04-25
Published Online: 2025-05-23
Published in Print: 2025-11-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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  2. Bounding the degree of generic sharp transitivity
  3. Structure of an exotic 2-local subgroup in 𝐸7(𝑞)
  4. Set-stabilizers in solvable permutation groups
  5. Regular saturated formations of finite soluble groups
  6. On the 𝑝-length and 𝔘-class of a 𝑝-solvable finite group
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  8. Finite class 2 nilpotent and Heisenberg groups
  9. Minimal degrees for faithful permutation representations of groups of order 𝑝5 where 𝑝 is an odd prime
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  11. On the Grossman representations of the automorphism groups of free groups
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  13. The proper geometric dimension of the mapping class group of an orientable surface with punctures
  14. A note on complex hyperbolic lattices and strict hyperbolization
https://doi.org/10.1515/jgth-2024-0190

Articles in the same Issue

  1. Frontmatter
  2. Bounding the degree of generic sharp transitivity
  3. Structure of an exotic 2-local subgroup in 𝐸7(𝑞)
  4. Set-stabilizers in solvable permutation groups
  5. Regular saturated formations of finite soluble groups
  6. On the 𝑝-length and 𝔘-class of a 𝑝-solvable finite group
  7. Finite 2-groups with exactly three automorphism orbits
  8. Finite class 2 nilpotent and Heisenberg groups
  9. Minimal degrees for faithful permutation representations of groups of order 𝑝5 where 𝑝 is an odd prime
  10. 𝜏-Tilting finiteness of group algebras of semidirect products of abelian 𝑝-groups and abelian 𝑝′-groups
  11. On the Grossman representations of the automorphism groups of free groups
  12. Strong indecomposability of the outer automorphism groups of nonabelian free profinite groups
  13. The proper geometric dimension of the mapping class group of an orientable surface with punctures
  14. A note on complex hyperbolic lattices and strict hyperbolization
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