Free and discrete subgroups generated by two quaternionic Heisenberg translations

TL;DR

The paper analyzes when the subgroup generated by two quaternionic Heisenberg translations in ${\rm Sp}(2,1)$ is discrete and free. It develops a constructive criterion via Klein's combination theorem by comparing isometric Cygan spheres and establishing containment bounds in the quaternionic Korányi framework, yielding explicit inequalities involving $K(p_i)$, $\zeta_i$, and $v_i$. The main theorem specializes to Parker--Kalane in the complex (``$\mathbb{C}\times\mathbb{R}$'') setting and extends prior vertical-translation results to the quaternionic setting. This provides a practical, geometric method to produce free, discrete two-generator subgroups in quaternionic hyperbolic space with potential applications to the broader theory of discrete groups in rank-one symmetric spaces.

Abstract

Let $A$ and $B$ be two Heisenberg translations of ${\rm Sp}(2,1)$ with distinct fixed points. We provide sufficient conditions that guarantee the subgroup $\langle A, B \rangle $ is discrete and free by using Klein's combination theorem.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 4.1
• Proposition 4.2
• proof