Quaternionic hyperbolic Fenchel–Nielsen coordinates

Original Paper


Let \(\mathrm {Sp}(2,1)\) be the isometry group of the quaternionic hyperbolic plane \(\mathbf{H}_{\mathbb {H}}^2\). An element g in \(\mathrm {Sp}(2,1)\) is hyperbolic if it fixes exactly two points on the boundary of \(\mathbf{H}_{\mathbb {H}}^2\). We classify pairs of hyperbolic elements in \(\mathrm {Sp}(2,1)\) up to conjugation. A hyperbolic element of \(\mathrm {Sp}(2,1)\) is called loxodromic if it has no real eigenvalue. We show that the set of \(\mathrm {Sp}(2,1)\) conjugation orbits of irreducible loxodromic pairs is a \((\mathbb {C}\mathbb {P}^1)^4\) bundle over a topological space that is locally a semi-analytic subspace of \(\mathbb {R}^{13}\). We use the above classification to show that conjugation orbits of ‘geometric’ representations of a closed surface group (of genus \(g \ge 2\)) into \(\mathrm {Sp}(2,1)\) can be determined by a system of \(42g-42\) real parameters. Further, we consider the groups \(\mathrm {Sp}(1,1)\) and \(\mathrm {GL}(2, \mathbb {H})\). These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of ‘geometric’ surface group representations into these groups.


Hyperbolic space Quaternions Free group representations Character variety Loxodromic 

Mathematics Subject Classification (2010)

Primary 57M50 Secondary 51M10 20H10 30F40 15B33 



We thank Giannis Platis and Wensheng Cao for many comments and suggestions on a first draft of this paper. We are grateful to the referee for writing an elaborate report on our paper and for many suggestions to improve the structure of the paper. A part of this work was carried out when one of the authors, Gongopadhyay, was visiting the UNSW Sydney supported by the Indo-Australia EMCR Fellowship of the Indian National Science Academy (INSA): thanks to UNSW for hospitality and INSA for the fellowship during the visit. Thanks are also due to Anne Thomas for organising a seminar by Gongopadhyay on this topic at the Sydney University. Kalane thanks a UGC research fellowship for supporting him through out this project. Gongopadhyay acknowledges partial support from SERB-DST MATRICS project: MTR/2017/000355.


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Authors and Affiliations

  1. 1.Indian Institute of Science Education and Research (IISER) MohaliS.A.S. NagarIndia

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