An Irrational-slope Thompson's Group

José Burillo, Brita Nucinkis, Lawrence Reeves

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The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_\tau$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of $F$ in $F_\tau$ are undistorted.
Original languageEnglish
Pages (from-to)809-839
Number of pages31
JournalPublicacions Matemàtiques
Issue number2
Publication statusPublished - 1 Jun 2021


  • math.GR
  • 20F65

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