Rank gradient, cost of groups and the rank versus Heegaard genus problem

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.

References in corpus (1)

• Dimension and randomness in groups acting on rooted trees

Cited by in corpus (9)

• A survey of Measured Group Theory
• Rank gradient and torsion groups
• Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds
• Rank and genus of 3-manifolds
• Expander Graphs in Pure and Applied Mathematics
• A $p$-group with positive Rank Gradient
• Heegaard splittings and 1-relator groups
• Rank gradient in co-final towers of certain Kleinian groups
• On the doubled tetrus