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New York Journal of Mathematics 6 (2000), 107-117.

Test Elements and the Retract Theorem in Hyperbolic Groups

John C. O'Neill and Edward C. Turner

Published: July 21, 2000
Keywords: test words, test elements, endomorphisms, automorphisms, hyperbolic groups
Subject: 20E36, 20F232


We prove that in many, perhaps all, torsion free hyperbolic groups, test elements are precisely those elements not contained in proper retracts. We also show that all Fuchsian groups have this property. Finally, we show that all surface groups except ${\mathbb Z} \times {\mathbb Z}$ have test elements.

Author information:
John C. O'Neill :
DFMS, United States Air Force Academy, CO 80840-6252

Edward C. Turner :
Dept. of Math., University at Albany, Albany, NY 12222