Elliptic triangle groups in PU(2,1), Lagrangian triples and momentum maps

Abstract

We determine the possible eigenvalues of elliptic matrices $A,B,C$ in $PU(2,1)$ satisfying $ABC=1$. This is done by describing geometrically the image of a group-valued momentum map for the (non-compact) group action of $PU(2,1)$ by conjugation on $C_1\times C_2$ where $C_1$ and $C_2$ are fixed elliptic conjugacy classes in $PU(2,1)$. Contrary to the compact case, this image is not always convex; rather it is the union of one, two or three convex polygons in $T^2/S_2$. The main motivation was to analyze elliptic triangle groups in $PU(2,1)$ such as Mostow’s lattices.