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INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY
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 Title & Authors
INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY
Kim, Hong Chan;
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 Abstract
The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere (0, 3), a one-holed torus (1, 1), and a four-holed sphere (0, 4). For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product <,> which gives the information about locations of axes of loxodromic elements. The signs of traces of holonomy elements, which are calculated by lifting a representation from PSL(2, ) to SL(2, ), play a very important role in determining the discreteness of holonomy groups.
 Keywords
Fricke space;involution;discrete holonomy group;
 Language
English
 Cited by
 References
1.
L. Bers and F. Gardiner, Fricke spaces, Adv. in Math. 62 (1986), no. 3, 249-284. crossref(new window)

2.
M. Culler, Lifting representations to covering groups, Adv. in Math. 59 (1986), no. 1, 64-70. crossref(new window)

3.
W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200-225. crossref(new window)

4.
W. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557-607. crossref(new window)

5.
W. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791-845.

6.
W. Goldman, Locally homogeneous geometric manifolds, Proceedings of the International Congress of Mathematicians. Volume II, 717-744, Hindustan Book Agency, New Delhi, 2010.

7.
H. C. Kim, Embedding of the Teichmuller Space into Goldman Space, J. Korean Math. Soc. 43 (2006), no. 6, 1231-1252. crossref(new window)

8.
N. Kuiper, On convex locally projective spaces, Convegno Internazionale di Geometria Differenziale, Italia, 1953, pp. 200-213. Edizioni Cremonese, Roma, 1954.

9.
W. Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), no. 1, 91-103. crossref(new window)

10.
K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Science Publications, Oxford University Press, 1998.

11.
J. Morgan and P. Shalen, Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston's compactness theorem, Ann. of Math. (2) 127 (1988), no. 3, 457-519. crossref(new window)

12.
C. Procesi, The invariant theory of n ${\times}$ n matrices, Adv. in Math. 19 (1976), no. 3, 306-381. crossref(new window)

13.
J. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer, 1994.

14.
D. Sullivan and W. Thurston, Manifolds with canonical coordinates: some examples, Enseign. Math. 29 (1983), no. 1-2, 15-25.

15.
A. Weil, On discrete subgroups of Lie groups I, Ann. of Math. (2) 72 (1960), 369-384. crossref(new window)