Quasi-isometric rigidity of extended admissible groups

Abstract

We introduce the class of extended admissible groups, which include both fundamental groups of non-geometric 3-manifolds and Croke-Kleiner admissible groups. We show that the class of extended admissible groups is quasi-isometrically rigid.

Figures (2)

• Figure 1: A diagram of the path $\gamma$.
• Figure 2: The picture illustrates two fibers $F_{S(k)}^{-}$ and $F_{k}^{+}$ of $E_k$ (resp. $F^{+}_{S(k)}$ and $F^{-}_{S^2(k)}$ of $E_{S(k)}$) intersecting at $x_k$ (resp. $x_{S(k)}$). Here $F_{S(k)}^{-}$ is the $S(k)$-fiber of $E_k$ closest to $E_{S(k)}$ and $F_{S(k)}^{+}$ is the $S(k)$-fiber of $E_{S(k)}$ closest to $E_{k}$. The path $\gamma$ intersects $F_{S(k)}^{-}$ and leaves $F_{S(k)}^{-}$ at $\gamma(t_k)$.

Theorems & Definitions (131)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.5
• Corollary 1.6
• Corollary 1.7
• Definition 2.1: Quasi-action
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: Section 2.5 of CM17