# Faithful real representations of groups of $F$-type

Document Type: Research Paper

Authors

1 Department of Mathematics Fairfield Universiry Fairfield, CT 06840

2 Department of Mathematics University of Hamburg

3 Department of Mathematics University of Hmaburg

Abstract

‎Groups of $F$-type were introduced in [B‎. ‎Fine and G‎. ‎Rosenberger‎, ‎Generalizing Algebraic Properties of Fuchsian Groups‎, \emph{London Math. Soc. Lecture Note Ser.}, \textbf{159} (1991)‎ ‎124--147.] as a natural algebraic generalization of Fuchsian groups‎. ‎They can be considered as the analogs of cyclically pinched one-relator groups where torsion is allowed‎. ‎Using the methods In [B‎. ‎Fine‎. ‎M‎. ‎Kreuzer and G‎. ‎Rosenberger‎, ‎Faithful Real Representations of Cyclically Pinched One-Relator Groups, Int. J. Group Theory, 3 ‎(2014) 1--8.] we prove that any hyperbolic group of $F$-type has a faithful representation in $PSL(2,\mathbb R)$‎. ‎From this we also obtain that a cyclically pinched one-relator group has a faithful real representation if and only if it is hyperbolic‎. ‎We further survey the many nice properties of groups of $F$-type‎. ‎

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