Affiliation:
1. Department of Pure Mathematics and Mathematical Statistics University of Cambridge, Centre for Mathematical Sciences Cambridge UK
2. Institut für Geometrie TU Dresden, Faculty of Mathematics Dresden Germany
Abstract
AbstractWe show that for any finite‐rank–free group , any word‐equation in one variable of length with constants in fails to be satisfied by some element of of word‐length . By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group . Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including for all , and the fundamental groups of all closed hyperbolic surfaces and 3‐manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group and a sequence of word‐equations with constants in for which every nonsolution in is of word‐length strictly greater than logarithmic.
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