bibkey: grunewald2015coverhomology
authors: Fritz Grunewald; Michael Larsen; Alexander Lubotzky; Justin Malestein
year: 2015
title: Arithmetic quotients of the mapping class group
doi: 10.1007/s00039-015-0352-5
url: https://arxiv.org/abs/1307.2593v2
claim: Cover-homology and finite-group representation data provide virtual arithmetic quotients; this does not supply a prescribed global mapping-class quotient factorization.
strata_touched: []
license: citation-only
triage: anchor

Cover homology and virtual arithmetic quotients

Verified locator

Fritz Grunewald, Michael Larsen, Alexander Lubotzky and Justin Malestein, Arithmetic quotients of the mapping class group, Geometric and Functional Analysis (2015). DOI: 10.1007/s00039-015-0352-5. https://doi.org/10.1007/s00039-015-0352-5 https://arxiv.org/abs/1307.2593v2

Source scope

The paper associates arithmetic groups to rational irreducible representations of finite groups. When the finite group can be generated by fewer than g elements, the associated arithmetic group is a virtual quotient of Mod(Sigma_g). Finite-index subgroups on both sides are part of the scope; a prescribed quotient of the whole mapping-class group does not follow.

Correspondence to the theory

Sections 27-32 of SURFACE_CONGRUENCE_OBSERVERS fix a characteristic finite-index surface subgroup N. For centerless S=pi/N and trivial induced outer action, removing the unique inner base action gives a corrected representation Psi on H1(N,F_l). If C is the principal outer kernel for pi/[N,N]N^l and K=ker(Psi), then K/C embeds in H1(S,H1(N,F_l)). Equality holds when l does not divide |S|. A specified nonabelian finite simple target factors through the outer action exactly when it factors through Psi.

The relative Frattini argument identifies an l-group kernel in the outer action on Q/Phi(P), for a normal l-subgroup P of a finite Q. Thus deeper l-group refinements over the same N have the same factorization ability toward that simple target as the first mod-l homology layer. Under the stated base hypotheses, a finite joint family at several primes succeeds only if one first homology layer succeeds. These particular implications are derived in the volume, not attributed to a numbered theorem of the cited paper.

Boundaries

A virtual arithmetic image does not identify the kernel of a separately specified quantum representation. The finite factorization test must use paired images of the same mapping-class generators in the fixed target and in Psi. It does not itself produce a successful cover or prime.

The conclusions retain the centerless, outer-trivial base hypotheses and the same source homomorphism throughout. They neither equate all outer kernels nor assert that arbitrary finite quotients admit these refinements, and they do not resolve the general curve-orbit congruence conjecture.