Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2022; 59(2): 351-360

Published online March 31, 2022 https://doi.org/10.4134/BKMS.b210150

Copyright © The Korean Mathematical Society.

Rigidity of a rank $1$ cusp of punctured-surface groups in hyperbolic $4$-space

Youngju Kim

Konkuk University

Abstract

We prove that a punctured-torus group of hyperbolic $4$-space which keeps an embedded hyperbolic $2$-plane invariant has a strictly parabolic commutator. More generally, this rigidity persists for a punctured-surface group.

Keywords: Hyperbolic geometry, hyperbolic $4$-space, parabolic isometry, punctured-surface group, punctured-torus group, deformation, rigidity

MSC numbers: Primary 57M50, 51M09; Secondary 30F40, 22E40

Supported by: This paper was supported by Konkuk University in 2018.