Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space

Abstract

We study representations of fundamental groups of closed orientable surfaces into \(\mathrm{SL}(2,\mathbb{H})\), whose projectivization \(\mathrm{PSL}(2,\mathbb{H})\) is the group of orientation-preserving isometries of real hyperbolic \(5\)-space, with the aim of developing a quaternionic analogue of Fenchel--Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is \(15\)-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there. For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally \(6\)-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus \(g\), this yields \(30g-30\) real parameters.