Reversible quaternionic hyperbolic isometries

Abstract

Let G be a group. An element g ∈ G is called reversible if it is conjugate to g −1 within G, and called strongly reversible if it is conjugate to g −1 by an order two element of G. Let H n H be the n-dimensional quaternionic hyperbolic space. Let PSp(n, 1) be the isometry group of H n H . In this paper, we classify reversible and strongly reversible elements in Sp(n) and Sp(n, 1). Also, we prove that all the elements of PSp(n, 1) are strongly reversible.

Reversibility of affine transformations

“…Investigation of reversible and strongly reversible elements in a group is an active area of current research; see [11] for an elaborate exposition of this theme from the geometric point of view. A complete classification of reversible and strongly reversible elements is not available in the literature except for the case of a few families of infinite groups, which include the compact Lie groups, real rank one classical groups and isometry groups of hermitian spaces; see [2,5,11]. In this article, by reversibility in a group G, we mean a classification of reversible and strongly reversible elements in G.…”

Reversibility of Hermitian Isometries

“…91]. Bhunia and Gongopadhyay have given a solution to this problem in [BG,Theorem 1.2]. However, the proof in [BG] is geometric and uses the notion of 'projective points'.…”