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Overcommuting pairs in groups and 3-manifolds bounding them

Liechti, Livio ; Marché, Julien

In: Journal of the London Mathematical Society, 2020, p. -

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL2, related to the Steinberg relation.

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Minimal pseudo-Anosov stretch factors on nonoriented surfaces

Liechti, Livio ; Strenner, Balázs

In: Algebraic & Geometric Topology, 2020, vol. 20, no. 1, p. 451–485

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliations on the closed orientable surfaces of genus 1, 3, 5, 7, 9 and 11. As a ...

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On the genus defect of positive braid knots

Liechti, Livio

In: Algebraic & Geometric Topology, 2020, vol. 20, no. 1, p. 403–428

We show that the difference between the Seifert genus and the topological 4–genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a genus difference less than any fixed constant is characterised by finitely...

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