An ascending HNN extension of a free group inside SL(2,C)

Details An ascending HNN extension of a free group inside SL(2,C) An ascending HNN extension of a free group inside SL(2,C)

2004-12-07

arxiv.org/abs/math/0412136v2

Proc. Amer. Math. Soc. 134 (2006), no. 11, 3131-3136

Mathematics

Group Theory

7 pages. V2: minor improvements in exposition

Scientific paper

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. The main ingredient in our construction is a specific finite volume (noncompact) 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 its fundamental group 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.

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