scholarly journals LOCAL COORDINATES FOR COMPLEX AND QUATERNIONIC HYPERBOLIC PAIRS

2021 ◽  
pp. 1-22
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
SAGAR B. KALANE
Keyword(s):  
Fundamental Group ◽  
Arbitrary Dimension ◽  
Local Dimension ◽  
Oriented Surface ◽  
Local Coordinates ◽  
Local Parametrization

Abstract Let $G(n)={\textrm {Sp}}(n,1)$ or ${\textrm {SU}}(n,1)$ . We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$ . Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for ${\textrm {SU}}(3,1)$ . We extend this notion and classify $G(n)$ -conjugation orbits of such elements in arbitrary dimension. For $n=3$ , they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus $g \geq 2$ ) oriented surface into $G(3)$ .

scholarly journals INDUCED QUANTUM GRAVITY ON A RIEMANN SURFACE

2003 ◽  
Vol 18 (24) ◽  
pp. 4371-4401 ◽  
Author(s):  
G. BANDELLONI ◽  
S. LAZZARINI
Keyword(s):  
Riemann Surface ◽  
Simple Model ◽  
Quantum Gravity ◽  
Arbitrary Dimension ◽  
Perturbative Expansion ◽  
Target Space ◽  
Quantum Corrections ◽  
Classical Formula ◽  
Local Coordinates ◽  
Nontrivial Character

Induced quantum gravity dynamics built over a Riemann surface is studied in arbitrary dimension. Local coordinates on the target space are given by means of the Laguerre–Forsyth construction. A simple model is proposed and perturbatively quantized. In doing so, the classical [Formula: see text]-symmetry turns out to be preserved on-shell at any order of the ℏ perturbative expansion. As a main result, due to quantum corrections, the target coordinates acquire a nontrivial character.

scholarly journals Families of SU(2) representations for mapping cylinders of periodic monodromy

1997 ◽  
Vol 40 (2) ◽  
pp. 383-392
Author(s):  
G. Daskalopoulos ◽  
S. Dostoglou ◽  
R. Wentworth
Keyword(s):  
Fundamental Group ◽  
Fixed Points ◽  
Conjugacy Classes ◽  
Oriented Surface ◽  
3 Dimensional ◽  
Dimensional Mapping

We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.

scholarly journals Frobenius algebras derived from the Kauffman bracket skein algebra

2016 ◽  
Vol 25 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Charles Frohman ◽  
Nel Abdiel
Keyword(s):  
Fundamental Group ◽  
Central Extension ◽  
Function Field ◽  
Underlying Surface ◽  
Character Variety ◽  
Oriented Surface ◽  
Variable Formula ◽  
Kauffman Bracket ◽  
Frobenius Algebras ◽  
Character Ring

The Kauffman bracket skein algebra of a compact oriented surface when the variable [Formula: see text] in the Kauffman bracket is set equal to [Formula: see text], where [Formula: see text] is an odd counting number, is a central extension of the ring of [Formula: see text]-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

scholarly journals THE GENERALIZED DEHN TWIST ALONG A FIGURE EIGHT

2013 ◽  
Vol 05 (03) ◽  
pp. 271-295
Author(s):  
YUSUKE KUNO
Keyword(s):  
Fundamental Group ◽  
Group Ring ◽  
Boundary Component ◽  
Double Point ◽  
Dehn Twist ◽  
Oriented Surface ◽  
The Right

For any unoriented loop on a compact connected oriented surface with one boundary component, we introduce a generalized Dehn twist along the loop as a certain automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this corresponds to the right-handed Dehn twist, and in particular is realized as a diffeomorphism of the surface. We investigate the case where the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a diffeomorphism.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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