2005/09/02 by Marc Lackenby, Lackenby, Marc
Computer Science · Mathematics · #20E07 #20F65 #57N10 #94B05 #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR #math.GT #msc:20E07 #msc:20F65 #msc:57N10 #msc:94B05
paper · pdf · doi:10.48550/arxiv.math/0509036
35 pages, 2 figures; minor corrections and improvements from version 2
openalex publication_date 2005/09/02 · arxiv created 2005/12/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the homology of finite index subgroups Gi of a given finitely presented group G. Specifically, we examine dp(Gi), which is the dimension of the first homology of Gi, with mod p coefficients. We say that a collection of finite index subgroups Gi has linear growth of mod p homology if the infimum of dp(Gi)/[G:Gi] is positive. We show that if this holds and each Gi is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to Gi. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance.