Reconfiguration of square-tiled surfaces

Abstract: A square-tiled surface is a special case of a quadrangulation of a surface, that can be encoded as a pair of permutations in \(S_n \times S_n\) that generates a transitive subgroup of \(S_n\). Square-tiled surfaces can be classified into different strata according to the total angles around their conical singularities. Among other parameters, strata fix the genus and the size of the quadrangulation. Generating a random square-tiled surface in a fixed stratum is a widely open question. We propose a Markov chain approach using "shearing moves": a natural reconfiguration operation preserving the stratum of a square-tiled surface. In a subset of strata, we prove that this Markov chain is irreducible and has diameter \(O(n^2)\), where \(n\) is the number of squares in the quadrangulation.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( slides )

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Geometry and topology online

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The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.

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