# Faithful real representations of cyclically pinched one-relator groups

Document Type: Second Biennial International Group Theory Conference 2013

Authors

1 Fair eld University

2 Universitat Passau

3 University of Hamburg

Abstract

‎‎In [4,5] using faithful complex representations of cyclically pinched and conjugacy pinched one-relator groups we proved that any limit group has a faithful representation in $PSL(2,C)$‎. ‎Further this representation can be effectively constructed using the JSJ decomposition‎. ‎In this note we show that any hyperbolic cyclically pinched one-relator group with maximal amalgamated subgroups in each factor has a 2-dimensional faithful real representation‎.

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Main Subjects

### References

P. Ackermann, B. Fine and G. Rosenberger (2007). On Surface Groups: Motivating Examples in Combinatorial Group Theory. Groups St. Andrews 2005}, London Math. Soc. Lecture Notes Ser., Cambridge Univ. Press, Cambridge. 1 (339), 96-129
B. Baumslag (1967). Residually free groups. Proc. London Math. Soc. (3). 17, 402-418
G. Baumslag (1962). On generalized free products. Math. Z.. 78, 423-438
B. Fine and G. Rosenberger (2011). A note on Faithful Representations of Hyperbolic Limit Groups. Groups Complex. Cryptol.. 3 (2), 349-355
B. Fine and G. Rosenberger Faithful Representations of Limit Groups II. Groups Complex. Cryptol., to appear.
B. Fine and G. Rosenberger (1991). Generalizing Algebraic Properties of Fuchsian Groups. Groups-St. Andrews 1989, London Math. Soc., Lecture Note Ser., Cambridge Univ. Press, Cambridge. 1 (159), 124-147
O. Kharlamapovich and A. Myasnikov (1998). Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz. J. Algebra. 200, 472-516
O. Kharlamapovich and A. Myasnikov (1998). Affine varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups. J. Algebra. 200, 517-569
O. Kharlamapovich and A. Myasnikov (2005). Implicit function theorem over free groups. J. Algebra. 290, 1-203
O. Kharlamapovich and A. Myasnikov (2005). Effective JSJ decompositions. Contemp. Math.. 378, 87-212
O. Kharlamapovich and A. Myasnikov (2006). Elementary theory of free nonabelian groups. J. Algebra. 302, 451-552
O. Kharlamapovich and A. Myasnikov (1998). Hyperbolic groups and free constructions. Trans. Amer. Math. Soc.. 350 (2), 571-613
G. Rosenberger (1991). Linear representations of cyclically pinched one-relator groups. Siberian Math. J.. 32, 203-206
P. Shalen (1979). Linear representations of certain amalgamated products. J. Pure Appl. Algebra. 15, 187-197
A. S. Rapinchuk, V. V. Benyash-Krivetz and V. I. Chernousov (1996). Representation varieties of the fundamental groups of compact orientable surfaces. Israel J. Math.. 93, 29-71
Z. Sela (1995). The isomorphism problem for hyperbolic groups I.. Ann. of Math. (2). 141 (2), 217-283
Z. Sela (2001). Diophantine geometry over groups I: Makanin-Razborov diagrams. Publ. Math. Inst. Hautes Etudes Sci.. (93), 31-105
Z. Sela (2003). Diophantine geometry over groups II: Completions. closures and formal solutions, Israel J. Math.. 134, 173-254
Z. Sela (2005). Diophantine geometry over groups III: Rigid and solid solutions. Israel J. Math.. 147, 1-73
Z. Sela (2004). Diophantine geometry over groups IV: An iterative procedure for validation of a sentence. Israel J. Math.. 143, 1-130
Z. Sela (2005). Diophantine geometry over groups V: Quantifier elimination. Israel J. Math.. 150, 1-197