On discreteness of subgroups of quaternionic hyperbolic isometries

2018/10/01 by Gongopadhyay, Krishnendu, Mishra, Mukund Madhav, Tiwari, Devendra
#Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1810.00657

Abstract

Let \bf H_{\mathbb H}ⁿ denote the n-dimensional quaternionic hyperbolic space. The linear group \rmSp(n,1) acts by the isometries of \bf H_{\mathbb H}ⁿ. A subgroup G of \rm Sp(n,1) is called Zariski dense if it does not fix a point on \bf H_{\mathbb H}ⁿ ∪ ∂ \bf H_{\mathbb H}ⁿ and neither it preserves a totally geodesic subspace of \bf H_{\mathbb H}ⁿ. We prove that a Zariski dense subgroup G of \rm Sp(n,1) is discrete if for every loxodromic element g ∈ G the two generator subgroup ⟨ f, g f g^-1 ⟩ is discrete, where the generator f ∈ \rmSp(n,1) is certain fixed element not necessarily from G.

Related

• Test map and Discreteness in SL(2, \mathbb H)
• Complex and Quaternionic hyperbolic Kleinian groups with real trace fields
• Jorgensen's inequality for quaternionic hyperbolic n-space
• Reversible Quaternionic Hyperbolic Isometries
• Quaternionic hyperbolic Kleinian groups with commutative trace skew-fields