Integer points in higher Teichmueller spaces, arithmetic height and a finiteness result (Marc Burger, ETHZ)

01.10.2026 13:00 – 14:00

Loosely speaking, a higher Teichmueller space is a connected component H(G,g) of the G-character variety of a compact surface group of genus g formed of holonomy representations of special geometric structures related to the compact surface. For G=PSL(2,R) one recovers Teichmueller space while for G=PSL(3,R) one recovers proper convex projective structures. In this talk we define a notion of integer point in H(G,g) and two invariants thereof, the arithmetic dimension and the arithmetic height, which are both mapping class group invariants. The main result is the finiteness, up to mapping class group action, of the number of integer points for which both invariants are bounded. Interesting examples already abound for G=PSL(2,R) as then the notion of integer point encompasses the notion of semi-arithmetic group introduced by Schmutz-Schaller and Wolfart. We will exhibit an interesting class of integer points when G=SO(4,2), as in this case they come from maximal representations.
Joint work with Anna Wienhard.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Thursday 01.10.26, Séminaire "Groupes et géométrie"

entrée libre