Three-manifold subgroup growth, homology of coverings and simplicial volume
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.
scientific article; zbMATH DE number 1186651