University of IsfahanInternational Journal of Group Theory2251-76509320200901Faithful real representations of groups of $F$-type1431552375510.22108/ijgt.2018.112439.1494ENBenjaminFineDepartment of Mathematics
Fairfield Universiry
Fairfield, CT 06840AnjaMoldenhauerDepartment of Mathematics
University of HamburgGerhardRosnebergerDepartment of Mathematics
University of HmaburgJournal Article20170330Groups 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, <em>Int. J. Group Theory</em>, <strong>3</strong> (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. https://ijgt.ui.ac.ir/article_23755_2af5ed775b029833bc7eff5c527aaaeb.pdf