Bicycle tracks with hyperbolic monodromy – results and conjectures

by Prof. Gil Bor (CIMAT)

Wednesday May 13, 2026, 2:00 PM → 3:00 PM Europe/Paris

Salle Pellos, bât. 1R2 (IMT)

One associates with every closed plane curve its "bicycle monodromy" -- an element of the Mobius group $PSL(2,R)$, defined via a dynamical system associated with the curve, that keeps appearing in different guises for over 100 years. The Menzin Conjecture (1906) states that this monodromy is hyperbolic if the curve is simple and encloses an area more than π. It has been proven in 2006 under the assumption that the curve is convex. The general case is still open. I will present two recent related results: (1) another sufficient condition for hyperbolicity of the monodromy is that the curvature of the curve is bounded above pointwise by 1 (but is not identically 1). (2) A necessary condition for hyperbolicity is that the length of the curve is greater than 2π. The main tool for both results is a “hyperbolic development” interpretation of the bicycling monodromy of plane curves (rolling without slipping and twisting of the hyperbolic plane along the given curve). This is joint work with Luis Hernandez (CIMAT) and Sergei Tabachnikov (Penn State). https://arxiv.org/abs/2412.18676