Local rigidity of diagonally embedded triangle groups

Abstract: In studying moduli spaces of representations of surface groups, and more generally of hyperbolic groups, triangle groups are simple examples which can provide insight into the more general theory. Recent work of Alessandrini–Lee–Schaffhauser generalized the theory of higher Teichmüller spaces to the setting of orbifold surfaces, including triangle groups. In particular, they defined a "Hitchin component" of representations into $\mathrm{PGL}(n,\mathbb{R})$ which is homeomorphic to a ball and consists entirely of discrete and faithful representations. They compute the dimension of Hitchin components for triangle groups, and find that this dimension is positive except for a finite number of low-dimensional examples where the representations are rigid. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with a diagonal embedding into $\mathrm{PGL}(2n,\mathbb{R})$ or $\mathrm{PSp}(2n,\mathbb{R})$ is always locally rigid.

group theorygeometric topologymetric geometry

Audience: researchers in the topic

McGill geometric group theory seminar