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SUMMARY:Luke Jeffreys (University of Bristol)
DTSTART:20260205T133000Z
DTEND:20260205T143000Z
DTSTAMP:20260423T035542Z
UID:GaTO/133
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GaTO/133/">E
 uler characteristics of SL(2\,Z)-orbit graphs of square-tiled surfaces</a>
 \nby Luke Jeffreys (University of Bristol) as part of Geometry and topolog
 y online\n\nLecture held in Room B3.02 in the Zeeman Building\, University
  of Warwick.\n\nAbstract\nSquare-tiled surfaces (aptly named) are surfaces
  obtained by gluing together a collection of unit squares along their side
 s (the square torus being the simplest example). These surfaces are specia
 l cases of translation surfaces\, whose moduli space carries a natural act
 ion of the group $SL(2\,\\mathbb{R})$. The famous works of Eskin–Mirzakh
 ani and Eskin–Mirzakhani–Mohammadi were concerned with understanding t
 he orbits of this action.\n\nThis $SL(2\,\\mathbb{R})$-action restricts to
  an action of $SL(2\,\\mathbb{Z})$ on square-tiled surfaces\, and the orbi
 ts under this restricted action can be transformed into finite regular gra
 phs. It is a long-standing conjecture of McMullen that a specific family o
 f these orbit graphs in genus two forms a family of expander graphs. Provi
 ding indirect evidence for this conjecture\, I will describe joint work wi
 th Carlos Matheus in which we prove that the absolute values of the Euler 
 characteristics of the graphs in this family go to infinity (a requirement
  for any expander family). To do this\, we are required to count a variety
  of objects\, including integer points on algebraic hypersurfaces\, pseudo
 -Anosov homeomorphisms of fixed dilatation\, and orbifold points on certai
 n algebraic curves in moduli space.\n
LOCATION:https://researchseminars.org/talk/GaTO/133/
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