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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.

By Ashani Dasgupta

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