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\(SL(2)\) Representations of Finitely Presented Groups
G. W. Brumfiel and H. M. Hilden

Contemporary Mathematics
1995; 196 pp; softcover
Volume: 187
ISBN-10: 0-8218-0416-2
ISBN-13: 978-0-8218-0416-2
List Price: US$63
Member Price: US$50.40
Order Code: CONM/187
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This book is essentially self-contained and requires only a basic abstract algebra course as background. The book includes and extends much of the classical theory of \(SL(2)\) representations of groups.

Readers will find \(SL(2)\)Representations of Finitely Presented Groups relevant to geometric theory of three dimensional manifolds, representations of infinite groups, and invariant theory.

Features ...

  • A new finitely computable invariant \(H[\pi ]\) associated to groups and used to study the \(SL(2)\) representations of \(\pi\).
  • Invariant theory and knot theory related through \(SL(2)\) representations of knot groups.


Researchers in invariant theory, representation theory of infinite groups, and applications of group representation theory to low dimensional topology.


"A useful algebraic framework for students and researchers concerned with representation spaces. Apart from the contribution it offers to the algebraic insight in the topic it should be helpful especially for topologists who look at knots or 3-manifolds from the Culler-Shalen perspective ... the monographis of considerable merit."

-- Zentralblatt MATH

Table of Contents

  • The definition and some basic properties of the algebra \(H[\pi ]\)
  • A decomposition of the algebra \(H[\pi ]\) when \(\frac 12\in k\)
  • Structure of the algebra \(H[\pi ]\) for two-generator groups
  • Absolutely irreducible \(SL(2)\) representations of two-generator groups
  • Further identities in the algebra \(H[\pi ]\) when \(\frac 12\in k\)
  • Structure of \(H^+[\pi _n]\) for free groups \(\pi _n\)
  • Quaternion algebra localizations of \(H[\pi ]\) and absolutely irreducible \(SL(2)\) representations
  • Algebro-geometric interpretation of \(SL(2)\) representations of groups
  • The universal matrix representation of the algebra \(H[\pi ]\)
  • Some knot invariants derived from the algebra \(H[\pi ]\)
  • Appendix A
  • Appendix B
  • References
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