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

We determine the possible eigenvalues of elliptic matrices A,B,CA,B,C in PU(2,1)PU(2,1) satisfying ABC=1ABC=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)PU(2,1) by conjugation on C1×C2C1×C2 where C1C1 and C2C2 are fixed elliptic conjugacy classes in PU(2,1)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 T2/S2T2/S2. The main motivation was to analyze elliptic triangle groups in PU(2,1)PU(2,1) such as Mostow’s lattices.