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

Miklós Abért, Nikolay Nikolov (2012)

Journal of the European Mathematical Society

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.

Regenerating hyperbolic cone 3-manifolds from dimension 2

Joan Porti (2013)

Annales de l’institut Fourier

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regenerations of the polygon, provided that the perimeter is minimal.

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