Heegaard genus and property τ for hyperbolic 3-manifolds

D. D. Long, Alexander Lubotzky, Alan W. Reid

Type: Article

Publication Date: 2007-10-25

Citations: 26

DOI: https://doi.org/10.1112/jtopol/jtm010

Abstract

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property τ. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

Locations

arXiv (Cornell University) - PDF
Journal of Topology - View

Similar Works

Action	Title	Year	Authors
+	Heegaard genus and Property 'tau' for hyperbolic 3-manifolds	2007	D. D. Long Alexander Lubotzky Alan W. Reid
+ PDF Chat	Rank gradient in co-final towers of certain Kleinian groups	2014	Darlan Girão
+	Rank gradient in co-final towers of certain Kleinian groups	2011	Darlan Girão
+	Rank gradient in co-final towers of certain Kleinian groups	2011	Darlan Girão
+ PDF Chat	Free subgroups of 3-manifold groups	2020	Mikhail Belolipetsky Cayo Dória
+	Kleinian Groups and Hyperbolic 3-Manifolds	2003
+	The Heegaard genus of hyperbolic 3-manifolds of small volume	1996	Andreĭ Vesnin Alexander Mednykh
+ PDF Chat	A generic Margulis number for hyperbolic 3-manifolds	2011	Peter B. Shalen
+	A Generic Margulis Number for Hyperbolic 3-Manifolds	2010	Peter B. Shalen
+	Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3–manifolds	2007	Juan Souto
+ PDF Chat	Short geodesics in hyperbolic 3–manifolds	2011	William Breslin
+	Heegaard genera in congruence towers of hyperbolic 3-manifolds	2011	BoGwang Jeon
+	A Crash Course on Kleinian Groups	2005	Caroline Series
+ PDF Chat	Heegaard and Regular Genus of 3-Manifolds	1995	Paola Cristofori Carlo Gagliardi Luigi Grasselli
+ PDF Chat	Gradients de Heegaard sous-logarithmiques d’une variété hyperbolique de dimension trois et fibres virtuelles	2011	Claire Renard
+ PDF Chat	Heegaard genera in congruence towers of hyperbolic 3-manifolds	2012	BoGwang Jeon
+ PDF Chat	Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds	2020	Tejas Kalelkar
+	Equivariant Heegaard genus of reducible 3-manifolds	2020	Scott A. Taylor
+	Heegaard splittings, the virtually Haken conjecture and Property tau	2002	Marc Lackenby
+	Topological symmetry groups of graphs in 3-manifolds	2012	Erica Flapan Harry Tamvakis

Works That Cite This (26)

Action	Title	Year	Authors
+ PDF Chat	Expansion in SL d (Z/q Z), q arbitrary	2011	Jean Bourgain Péter P. Varjú
+	Rank gradient, cost of groups and the rank versus Heegaard genus problem	2007	Miklós Abért Nikolay Nikolov
+	Spectral geometry, link complements and surgery diagrams	2008	Marc Lackenby
+	3-Manifold Groups	2015	Matthias Aschenbrenner Stefan Friedl Henry Wilton
+ PDF Chat	Immersed turnovers in hyperbolic 3-orbifolds	2010	Shawn Rafalski
+	On the doubled tetrus	2009	Jason DeBlois
+	On the length of nonsolutions to equations with constants in some linear groups	2024	Henry Bradford Jakob Schneider Andreas Thom
+ PDF Chat	Finite Covering Spaces of 3-manifolds	2011	Marc Lackenby
+ PDF Chat	Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property	2022	Martin R. Bridson A. W. Reid
+ PDF Chat	Free subgroups of 3-manifold groups	2020	Mikhail Belolipetsky Cayo Dória

Works Cited by This (20)

Action	Title	Year	Authors
+ PDF Chat	Arithmetic of Hyperbolic Manifolds	1992	Colin M. MacLachlan Alan W. Reid
+ PDF Chat	Simple quotients of hyperbolic 3-manifold groups	1998	D. D. Long Alan W. Reid
+ PDF Chat	Growth and generation in SL<sub>2</sub>(ℤ∕pℤ)	2008	H. A. Helfgott
+	Ramanujan duals II	1991	M. Burger Peter Sarnak
+ PDF Chat	A sum-product estimate in finite fields, and applications	2004	Jean Bourgain Nets Hawk Katz Terence Tao
+	Rigidity of cusps in deformations of hyperbolic 3-orbifolds	1993	Walter D. Neumann Alan W. Reid
+	Non-arithmetic groups in lobachevsky spaces	1987	M. Gromov Ilya Piatetski-Shapiro
+	ESTIMATES FOR THE NUMBER OF SUMS AND PRODUCTS AND FOR EXPONENTIAL SUMS IN FIELDS OF PRIME ORDER	2006	Jean Bourgain A. A. Glibichuk Sergeĭ Konyagin
+	Circle packings and co-compact extensions of Kleinian groups	1986	Robert Brooks
+	On groups of polynomial subgroup growth	1991	Alexander Lubotzky Avinoam Mann