[Paper Review] Congruence RFRS towers

Ian Agol, Matthew Stover|arXiv (Cornell University)|Dec 21, 2019

Geometric and Algebraic Topology29 references4 citations

TL;DR

This paper establishes a criterion for real or complex hyperbolic lattices to admit RFRS (residually finite rational solvable) towers composed entirely of congruence subgroups. Using this criterion, it proves that certain Bianchi groups PSL₂(𝒪𝒹) with d ≢ −1 mod 8 and square-free d are virtually fibered over congruence subgroups, and constructs the first examples of RFRS Kähler groups not isomorphic to subgroups of products of surface groups and abelian groups, via congruence subgroups of Deligne–Mostow lattices in PU(2,1).

ABSTRACT

We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.

Motivation & Objective

• To develop a criterion for hyperbolic lattices to admit RFRS towers consisting entirely of congruence subgroups.
• To establish that certain Bianchi groups PSL₂(𝒪𝒹) with d ≢ −1 mod 8 and square-free d are virtually fibered over congruence covers.
• To construct the first examples of RFRS Kähler groups that are not subgroups of products of surface groups and abelian groups.
• To extend the method of RFRS tower construction to congruence subgroups of arithmetic lattices in SO(n,1) and SU(n,1), particularly in the context of number fields and p-adic Bruhat–Tits buildings.

Proposed method

• Leverage the fact that Bianchi groups virtually embed into O(4,1;ℤ), enabling the use of congruence subgroups in this orthogonal group to construct RFRS towers.
• Apply a general construction using the commensurator of a lattice and the structure of the 𝔭-adic Bruhat–Tits building to inductively define group elements gₙ such that the intersection of conjugates forms a RFRS tower.
• Ensure that the quotient Γ / (Γ ∩ g₁Γg₁⁻¹) is an elementary abelian p-group by selecting g₁ appropriately in 𝒢(k), relying on the absence of p-torsion in H¹(Γ;ℤ).
• Use Reidemeister–Schreier algorithms implemented in Magma to compute presentations and abelianizations of principal congruence subgroups Γ(𝔭) in Bianchi groups.
• Analyze the first homology group H¹(Γ(𝔭);ℤ) to detect torsion and rank, using explicit group presentations and reduction modulo prime ideals 𝔭.
• Construct RFRS Kähler groups as congruence subgroups of Deligne–Mostow lattices in PU(2,1), proving they are not subgroups of products of surface groups and abelian groups.

Experimental results

Research questions

• RQ1 Under what conditions does a real or complex hyperbolic lattice admit an RFRS tower consisting entirely of congruence subgroups?
• RQ2 Do Bianchi groups PSL₂(𝒪𝒹) with d ≢ −1 mod 8 and square-free d virtually fiber over congruence subgroups?
• RQ3 Can one construct RFRS Kähler groups that are not subgroups of products of surface groups and abelian groups?
• RQ4 How can the RFRS tower construction be systematically applied to congruence subgroups of arithmetic lattices in SO(n,1) and SU(n,1) using p-adic Bruhat–Tits buildings?
• RQ5 What is the role of the first homology group’s p-torsion-free condition in enabling the inductive construction of RFRS towers?

Key findings

• The Bianchi groups PSL₂(𝒪𝒹) with d ≢ −1 mod 8 and square-free d admit a RFRS tower consisting entirely of congruence subgroups, implying they are virtually fibered on a congruence cover.
• For K = ℚ(√−1), the principal congruence subgroup Γ(𝔭) at norm 49 has rank 825 and torsion size 7⁶ in H¹(Γ(𝔭);ℤ), confirming nontrivial homology structure.
• For K = ℚ(√−2), the congruence subgroup at norm 67 has torsion size 2²⁰⁰·3¹³⁵·239⁶⁶·271⁶⁶·647⁶⁶·727⁶⁸·38011⁶⁶·47917⁶⁸, demonstrating large torsion in homology.
• For K = ℚ(√−11), the congruence subgroup at norm 59 has torsion size 2³⁴⁸·3⁴⁰⁸·5⁵⁸·7¹¹⁶·11⁵⁸·17¹²⁰·19⁵⁸·31⁵⁸·59⁵⁸·199⁵⁸·233⁶⁰·5279⁶⁰·20341⁵⁸, showing extensive torsion in homology.
• The first example of an RFRS Kähler group not isomorphic to a subgroup of a product of surface groups and abelian groups is realized as a congruence subgroup of a Deligne–Mostow lattice in PU(2,1).
• The construction of the RFRS tower relies on the absence of p-torsion in H¹(Γ;ℤ), and the inductive choice of group elements via the 𝔭-adic Bruhat–Tits building ensures the tower is RFRS.