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Conjugacy and dynamics in Thompson’s groups

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Abstract

We give a unified solution to the conjugacy problem for Thompson’s groups \(F, \,T\), and \(V\). The solution uses “strand diagrams”, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson’s groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson’s groups, and we use this correspondence to investigate the dynamics of elements of \(F\). Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.

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Notes

  1. A weaker version of the Newman’s Diamond Lemma suffices: see Theorem 1 in [18].

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Acknowledgments

The authors would like to thank Ken Brown and Martin Kassabov for a very careful reading of this paper and many helpful remarks. The authors would also like to thank Stephen Pride for providing fundamental references and Collin Bleak, Matt Brin, Jörg Lehnert and Mark Sapir for helpful conversations. Finally, the authors would like to thank the referee for many helpful comments and suggestions. The first author gratefully acknowledges partial support from an NSF Postdoctoral Research Fellowship while he was at Texas A&M University. The second author gratefully acknowledges the Centre de Recerca Matemàtica (CRM) and its staff for the support received during the development of this work.

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Authors and Affiliations

  1. Mathematics Program, Bard College, Annandale-on-Hudson, NY, 12504, USA

    James Belk

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Francesco Matucci

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  1. James Belk
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  2. Francesco Matucci
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Correspondence to Francesco Matucci.

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This work is part of the second author’s Ph.D. thesis at Cornell University.

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Belk, J., Matucci, F. Conjugacy and dynamics in Thompson’s groups. Geom Dedicata 169, 239–261 (2014). https://doi.org/10.1007/s10711-013-9853-2

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