Foldings in relatively hyperbolic groups

Abstract

Carrier graphs of groups representing subgroups of a given relatively hyperbolic groups are introduced and a combination theorem for relatively quasi-convex subgroups is proven. Subsequently a theory of folds for such carrier graphs is introduced and finiteness results for subgroups of locally relatively quasiconvex relatively hyperbolic groups and Kleinian groups are established.

Figures (6)

• Figure 1: The path $\gamma=[g,x]\cup[x,hx]\cup[hx,h^2x]\cup\ldots\cup[h^Nx,h^Ng]$ cannot be far from $Y$.
• Figure 2: A $(G,\mathbb{P})$-carrier graph with non-trivial stars $C_1$ and $C_2$, trivial star $C_3$, and essential vertices $u$ and $v$.
• Figure 3: A move that removes a redundant essential edge
• Figure 4: Another move that removes a redundant essential edge
• Figure 5: A move that introduces an new peripheral star

Theorems & Definitions (86)

• Theorem 1
• Theorem 2
• Definition 1.1: Osin Osin2006, Definition 2.35
• Definition 1.2
• Lemma 1.3: Osin Osin2006, Theorem 3.23 and Theorem 3.26
• Definition 1.4
• Definition 1.5: Osin Osin2006, Def. 4.9
• Definition 1.7
• Theorem 1.8
• Proof 1