Deformations of dihedral representations

Authors:
Michael Heusener and Eric Klassen

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3039-3047

MSC (1991):
Primary 57M25, 57M05

MathSciNet review:
1443155

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: G. Burde proved (1990) that the representation space of two-bridge knot groups is one-dimensional. The same holds for all torus knot groups. The aim of this note is to prove the following:

Given a knot we denote by its twofold branched covering space. Assume that there is a prime number such that . Then there exist representations of the knot group onto the binary dihedral group and these representations are smooth points on a one-dimensional curve of representations into .

**1.**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR**672956****2.**Gerhard Burde,*𝑆𝑈(2)-representation spaces for two-bridge knot groups*, Math. Ann.**288**(1990), no. 1, 103–119. MR**1070927**, 10.1007/BF01444524**3.**Gerhard Burde and Heiner Zieschang,*Knots*, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776****4.**D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen,*Plane curves associated to character varieties of 3-manifolds*, Invent. Math.**118**(1994), no. 1, 47–84. MR**1288467**, 10.1007/BF01231526**5.**Marc Culler and Peter B. Shalen,*Varieties of group representations and splittings of 3-manifolds*, Ann. of Math. (2)**117**(1983), no. 1, 109–146. MR**683804**, 10.2307/2006973**6.**Charles D. Frohman and Eric P. Klassen,*Deforming representations of knot groups in 𝑆𝑈(2)*, Comment. Math. Helv.**66**(1991), no. 3, 340–361. MR**1120651**, 10.1007/BF02566654**7.**Craig David Hodgson,*Degeneration and regeneration of geometric structures on three-manifolds*, Ph.D. thesis, Princeton University, 1986.**8.**Eric Paul Klassen,*Representations of knot groups in 𝑆𝑈(2)*, Trans. Amer. Math. Soc.**326**(1991), no. 2, 795–828. MR**1008696**, 10.1090/S0002-9947-1991-1008696-X**9.**R. C. Lyndon and J. L. Ullman,*Groups of elliptic linear fractional transformations*, Proc. Amer. Math. Soc.**18**(1967), 1119–1124. MR**0222182**, 10.1090/S0002-9939-1967-0222182-8**10.**Alexander Lubotzky and Andy R. Magid,*Varieties of representations of finitely generated groups*, Mem. Amer. Math. Soc.**58**(1985), no. 336, xi+117. MR**818915**, 10.1090/memo/0336**11.**André Weil,*Remarks on the cohomology of groups*, Ann. of Math. (2)**80**(1964), 149–157. MR**0169956**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57M25,
57M05

Retrieve articles in all journals with MSC (1991): 57M25, 57M05

Additional Information

**Michael Heusener**

Affiliation:
Uni–GH–Siegen Fachbereich Mathematik Hölderlinstraße 3 57068 Siegen Germany

Email:
heusener@hrz.uni-siegen.d400.de

**Eric Klassen**

Affiliation:
Department of Mathematics Florida State University Tallahassee Florida 32306

Email:
klassen@math.fsu.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04195-6

Keywords:
Knot groups,
group representations,
$\SU$

Received by editor(s):
September 7, 1993

Additional Notes:
The second author was supported in part by a National Science Foundation Postdoctoral Research Fellowship.

Communicated by:
Ronald Stern

Article copyright:
© Copyright 1997
American Mathematical Society