Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space
Abstract
We study representations of fundamental groups of closed orientable surfaces into SL(2,ℍ), whose projectivization PSL(2,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.
Related
- Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space
- Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space
- The computational ansatz for convex integration of hyperbolic systems and a resolution of the Strong Trace Conjecture
- Multi-target hyperbolic sieves and elliptic trace obstructions
- Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality
BibTeX
@misc{gongopadhyay2026trace,
title = {{Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space}},
author = {Krishnendu Gongopadhyay and Sagar B. Kalane and Abhishek Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11438},
eprint = {2609.11438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11438},
}
RIS
TY - UNPB
TI - Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space
AU - Krishnendu Gongopadhyay
AU - Sagar B. Kalane
AU - Abhishek Mukherjee
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11438
UR - https://arxiv.org/abs/2609.11438
AB - We study representations of fundamental groups of closed orientable surfaces into SL(2,ℍ), whose projectivization PSL(2,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.
ER -