Rips Machine

Let G be a finitely presented group acting minimally, stably and non trivially by isometries on an \( \mathbb{R} \) tree S. If G does not split over an arc stabilizer of S, then one of the following is true:

  1. There is a line \( L \subset S \) acted on by a subgroup H < G and \( N \triangleleft H \) the kernel of the action of H on S, so that H / N is virtually \( \mathbb{Z}^n \) for some n > 1.
  2. There is a closed hyperbolic cone 2-orbifold F and a normal subgroup \( N \triangleleft G \) with \( \pi_1 (F) \cong G/N \). Furthermore the action of G on S factors through \( \pi_1 (F) \).
  3. There is a finite graph of groups decomposition, \( \Gamma_1 \) of G with H < G a vertex group having the following properties.
    1. There is F, a cone 2-orbifold with boundary, and a normal subgroup \(N \triangleleft H \) with \( H/N \cong \pi_1 (F) \), so that the action of H on S factors through \( \pi_1(F) \)
    2. The edge groups of H in \( \Gamma \) correspond to the peripheral subgroups of F.
    3. All edge groups act trivially.

Published by Ashani Dasgupta

Pursuing Ph.D. in Geometric Group Theory at University of Wisconsin, Milwaukee

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