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Abért, Miklós; Nikolov, Nikolay

Rank gradient, cost of groups and the rank versus Heegaard genus problem. (English) Zbl 1271.57046

J. Eur. Math. Soc. (JEMS) 14, No. 5, 1657-1677 (2012).
The paper under review links two important problems, one in the field of low-dimensional topology and hyperbolic geometry, and one in the field of topological dynamics. The former is known as the Rank vs. Heegard genus Conjecture and asserts that for an orientable, compact \(3\)-dimensional hyperbolic manifold its Heegard genus equals the rank of its fundamental group. The latter problem asks whether every countable group has fixed price, i.e. whether every essentially free measure-preserving Borel action of such a group has the same cost. The core result of the paper (Theorem 2) sounds as follows: either the Rank vs. Heegaard genus conjecture is false or the fixed price problem has a negative solution.
Another important result of the paper is a formula (Theorem 1) relating the rank gradient of a Farber chain \((\Gamma_n)\) of subgroups of a finitely generated group \(\Gamma\) with the cost of the action of \(\Gamma\) on the boundary of the coset tree w.r.t. \((\Gamma_n)\). Theorem 1 is one of the main features leading to the core result of the paper.
Also Theorem 1 allows the authors to establish the fact that the rank gradient of a Farber chain in any residually finite group with an infinite amenable normal subgroup is zero (Theorem 3).
Reviewer: Alexander Kolpakov (Nashville)

Cited in 2 Reviews
Cited in 42 Documents

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
37B99 Topological dynamics

Keywords:

rank gradient; cost; Heegaard genus; fixed price
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References:

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