[Paper Review] Rank gradient, cost of groups and the rank versus Heegaard genus problem

Miklós Abért, Nikolay Nikolov|ArXiv.org|Jan 12, 2007

Geometric and Algebraic Topology15 references18 citations

TL;DR

This paper establishes a deep connection between the rank gradient of residually finite groups and the cost of their actions on boundary spaces of coset trees. It proves that the rank gradient equals the cost of the action minus one for Farber chains, and uses this to show that the Rank vs. Heegaard Genus Conjecture and the Fixed Price Problem in topological dynamics cannot both be true—offering strong evidence that the conjecture fails for arithmetic hyperbolic 3-manifolds.

ABSTRACT

We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in topological dynamics.

Motivation & Objective

• To investigate the relationship between rank gradient and cost in residually finite groups via Farber chains.
• To resolve the tension between the Rank vs. Heegaard Genus Conjecture and the Fixed Price Problem in topological dynamics.
• To demonstrate that if the Fixed Price Problem holds, then the ratio of Heegaard genus to rank in hyperbolic 3-manifolds can grow arbitrarily large.
• To show that for arithmetic lattices in SL₂(ℂ), the rank gradient vanishes along congruence subgroups, implying sublinear rank growth.
• To provide evidence that the Rank vs. Heegaard Genus Conjecture fails in the asymptotic regime of arithmetic hyperbolic 3-manifolds.

Proposed method

• Define the rank gradient of a group Γ with respect to a Farber chain (Γₙ) as the limit of (d(Γₙ)−1)/[Γ:Γₙ] as n→∞.
• Introduce the coset tree T(Γ, (Γₙ)) and its boundary ∂T, equipped with a Γ-action and normalized Haar measure.
• Establish that the rank gradient equals the cost of the action on ∂T minus one, i.e., RG(Γ, (Γₙ)) = cost(E)−1.
• Use Lackenby’s result on property (τ) and linear Heegaard genus growth in congruence towers of arithmetic lattices.
• Apply the cost–rank gradient identity to show that if cost is independent of the chain, then rank grows sublinearly relative to genus.
• Use the multiplicativity of cost-1 as a hypothesis to derive independence of rank gradient from the choice of Farber chain.

Experimental results

Research questions

• RQ1 Does the rank gradient of a residually finite group depend on the choice of Farber chain?
• RQ2 Can the Rank vs. Heegaard Genus Conjecture hold for hyperbolic 3-manifolds if the Fixed Price Problem has an affirmative solution?
• RQ3 Is the cost of a group action on a boundary space related to the asymptotic growth of the rank of finite-index subgroups?
• RQ4 Do arithmetic hyperbolic 3-manifolds exhibit arbitrarily large ratios of Heegaard genus to fundamental group rank?
• RQ5 Does the cost of a measurable action behave multiplicatively under passage to finite-index subgroups?

Key findings

• The rank gradient of a group with respect to a Farber chain equals the cost of the action on the boundary of the coset tree minus one.
• For arithmetic lattices such as SL₂(ℤ[i]), the rank gradient vanishes along congruence subgroups, implying cost 1 for the associated actions.
• If the Fixed Price Problem holds, then the ratio of Heegaard genus to rank in hyperbolic 3-manifolds can grow without bound.
• The counterexamples to the Rank vs. Heegaard Genus Conjecture are not exotic but arise generically in the asymptotic regime of arithmetic hyperbolic 3-manifolds.
• The result follows from combining Lackenby’s linear Heegaard genus growth in congruence towers with sublinear rank growth implied by zero rank gradient.
• The paper provides a negative answer to a question of Kechris and Miller on whether SL(2,D) has cost 1 iff D has infinitely many units, by showing cost 1 for such groups.