Reversible maps in the group of quaternionic Möbius transformations

Abstract

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations.

Surfaces containing two circles through each point

“…In what follows identify R 4 with the skew field H of quaternions, and R 3 with the set ImH of purely imaginary quaternions. Möbius transformations in R 4 are precisely the nondegenerate maps of the form q → (aq + b)(cq + d) −1 and q → (aq + b)(cq + d) −1 , where a, b, c, d ∈ H; see [16] for an exposition. Circles in R 4 are precisely the nondegenerate curves having a parametrization of the form α(u) = (au + b)(cu + d) −1 (outside one point), where a, b, c, d ∈ H are fixed and u ∈ R runs.…”

Reversible quaternionic hyperbolic isometries

“…[OS15,Theorem 6.11]. The reversible elements in PO o (n, 1) have been classified in [Gon11], and in [Sho08] using a different approach, and also see [LOS07]. We refer to [OS15,Chapter 6] for an extensive treatment of reversibility in Euclidean, spherical, and real hyperbolic geometries.…”

Reversible complex hyperbolic isometries

“…On the other hand from geometric point of view, the terminology 'reversible' is more commonly used, cf. [18,22,23,24]. We will use the terminology 'reversible' and 'strongly reversible'.…”

“…In higher dimensions, it follows from [13,Theorem 1.2] that every element of I(H n R ) is strongly reversible, also see [3,15,16,21,29]. The reversible elements in I o (H n R ) have been classified in [12,24], also see [18]. In [12], the first author obtained a linear-algebraic classification by identifying the orientation-preserving isometry group with SO o (n, 1).…”