Faithful Real Representations of Groups of $F$-type

Groups of F -type were introduced in [B. Fine and G. Rosenberger, Generalizing Algebraic Properties of Fuchsian Groups, London Math. Soc. Lecture Note Ser., 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,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.