An ascending HNN extension of a free group inside SL 2

Danny Calegari, Nathan M. Dunfield

Research output: Contribution to journalArticlepeer-review


We give an example of a subgroup of SL 2 ℂ which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL 2 ℂ of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005), The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of π 1 (M) in SL 2 ℂ with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group F we construct is actually free.

Original languageEnglish (US)
Pages (from-to)3131-3136
Number of pages6
JournalProceedings of the American Mathematical Society
Issue number11
StatePublished - Nov 2006
Externally publishedYes


  • Ascending HNN extension
  • Hyperbolic 3-manifold
  • SL ℂ

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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