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In this talk I will focus on Lyndon's equation, x^p y^q z^r=1, in tree-free groups. Tree-free groups are groups which act freely and without inversions by isometries on some Lambda-tree, where Lambda is an ordered abelian group.
Let G be a tree-free group and let x, y, z be elements in G. We show that if x^p y^q z^r =1 with integers p, q, r at least 4, then x, y and z commute.
As a result, the one-relator groups with x^p y^q z^r =1 as relator, are examples of hyperbolic and CAT(-1) groups which do not act freely on any Lambda-tree. This is joint work with N. Brady, A. Martino and S. O Rourke.
The New York Group Theory Seminar and some of the associated conferences are supported by funds from the National Science Foundation, Dean of Science, Maria Tamargo and Dean of Engineering, Joe Barba.