2017/08/21 by Gongopadhyay, Krishnendu, Kalane, Sagar B.
#15B33 #20H10 #30F40 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57M50 #Secondary 51M10
paper · doi:10.48550/arxiv.1708.06044
Let Sp(2,1) be the isometry group of the quaternionic hyperbolic plane \bf H\mathbb H2. An element g in Sp(2,1) is `hyperbolic' if it fixes exactly two points on the boundary of \bf H\mathbb H2. We classify pairs of hyperbolic elements in Sp(2,1) up to conjugation. A hyperbolic element of Sp(2,1) is called `loxodromic' if it has no real eigenvalue. We show that the set of Sp(2,1) conjugation orbits of irreducible loxodromic pairs is a (\mathbb C \mathbb P1)4-bundle over a topological space that is locally a semi-analytic subspace of \mathbb R13. We use the above classification to show that conjugation orbits of `geometric' representations of a closed surface group (of genus g ≥ 2) into Sp(2,1) can be determined by a system of 42g-42 real parameters. Further, we consider the groups Sp(1,1) and 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.