Three-manifold subgroup growth, homology of coverings and simplicial volume
type: Item
instance of: scholarly article
title: Three-manifold subgroup growth, homology of coverings and simplicial volume
zbMATH Open document ID: 0913.20017
author: Pieter Moree
author: Alexander G. Reznikov
published in: The Asian Journal of Mathematics
publication date: 1999-06-08T00:00:00Z
review text: This paper is concerned with the subgroup growth of 3-manifold groups. For a finitely generated group \(G\) let \(f(d)\) be the number of subgroups of index \(d\). It was conjectured by A. Lubotzky and A. Shalev that if \(G\) is the fundamental group of a hyperbolic 3-manifold, then there exists an absolute positive constant \(C\) such that, for infinitely many \(d\), \(f(d)>\exp(Cd)\). The authors show that for \(G\) as above \(f(d)>\exp(Cd^\alpha)\), for some positive constant \(\alpha\). In fact, they show this result for a priori a much wider class of 3-manifold groups, groups of manifolds that satisfy a certain richness condition (R), given the present status of Thurston's hyperbolization conjecture. The paper also contains a weak hyperbolization result for certain atoroidal 3-manifolds. A manifold can be considered hyperbolic in a weak sense if the Gromov simplicial volume is positive. It is shown that if \(M\) is an atoroidal 3-manifold and for infinitely many rational primes \(l\) there exists a rational prime \(p\), \(p=\pm 1\) modulo \(l\), and a surjective representation \(\rho_l\colon\pi_1(M)\to\text{SL}_2(F_q)\), where \(q\) is a power of \(p\), such that the covering of \(M\) corresponding to the kernel of \(\rho_l\) has trivial \(l\)-homology, then \(M\) has positive Gromov invariant.
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1186651
zbMATH Keywords: subgroup growth of 3-manifold groups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: hyperbolization conjecture
zbMATH Keywords: atoroidal 3-manifolds
zbMATH Keywords: Gromov simplicial volume
MaRDI profile type: Publication
DOI: 10.4310/AJM.1997.V1.N4.A6
description: scientific article; zbMATH DE number 1186651