Reversible Maps in Isometry Groups of Spherical, Euclidean and Hyperbolic Space

Short, Ian

doi:10.3318/pria.2008.108.1.33

Cited by 7 publications

(6 citation statements)

References 10 publications

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“…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'.…”

Section: Introductionmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

“…In [12], the first author obtained a linear-algebraic classification by identifying the orientation-preserving isometry group with SO o (n, 1). In [24], a geometric classification of the reversible elements in I o (H n R ) was obtained using the ball model of the hyperbolic space.…”

Section: Introductionmentioning

confidence: 99%

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Reversible complex hyperbolic isometries

Gongopadhyay

Parker

2013

Linear Algebra and its Applications

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“…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'.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reversible complex hyperbolic isometries

Gongopadhyay

Parker

2013

Linear Algebra and its Applications

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“…Using the ball model, Short [16] has studied the reality properties of conjugacy classes in the Möbius groups. The approach of Short is geometric.…”

Section: Introductionmentioning

confidence: 99%

Conjugacy classes in Möbius groups

Gongopadhyay

2010

Geom Dedicata

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Abstract. Let H n+1 denote the n + 1-dimensional (real) hyperbolic space. Let S n denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of S n is denoted by M (n). Let Mo(n) be its identity component which consists of all orientation-preserving elements in M (n). The conjugacy classification of isometries in Mo(n) depends on the conjugacy ofFor an element T in M (n), T and T −1 are conjugate in M (n), but they may not be conjugate in Mo(n). In the literature, T is called real if T is conjugate in Mo(n) to T −1 . In this paper we classify real elements in Mo(n).Let T be an element in Mo(n). Corresponding to T there is an associated element To in SO(n+1). If the complex conjugate eigenvalues of To are given by {e iθ j , e −iθ j }, 0 < θj ≤ π, j = 1, ..., k, then {θ1, ..., θ k } are called the rotation angles of T . If the rotation angles of T are distinct from eachother, then T is called a regular element. After classifying the real elements in Mo(n) we have parametrized the conjugacy classes of regular elements in Mo(n). In the parametrization, when T is not conjugate to T −1 , we have enlarged the group and have considered the conjugacy class of T in M (n). We prove that each such conjugacy class can be induced with a fibration structure.

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“…This notion is defined formally in §2; it has an analogous meaning to the real part of a complex number. [18]. In this paper we combine quaternion algebra with geometry, and in so doing we follow H. S. M. Coxeter ([8]), and P. G. Gormley ([11]-an extension of [8]).…”

Section: Introductionmentioning

confidence: 99%