By Lee Mosher

ISBN-10: 0821847120

ISBN-13: 9780821847121

This paper addresses questions of quasi-isometric tension and type for basic teams of finite graphs of teams, less than the belief that the Bass-Serre tree of the graph of teams has finite intensity. the most instance of a finite intensity graph of teams is one whose vertex and area teams are coarse Poincare duality teams. the most theorem says that, lower than convinced hypotheses, if $\mathcal{G}$ is a finite graph of coarse Poincare duality teams, then any finitely generated staff quasi-isometric to the elemental team of $\mathcal{G}$ can also be the basic workforce of a finite graph of coarse Poincare duality teams, and any quasi-isometry among such teams needs to coarsely guard the vertex and side areas in their Bass-Serre bushes of areas. in addition to a few uncomplicated normalization hypotheses, the most speculation is the "crossing graph condition", that is imposed on every one vertex crew $\mathcal{G}_v$ that is an $n$-dimensional coarse Poincare duality staff for which each incident facet workforce has confident codimension: the crossing graph of $\mathcal{G}_v$ is a graph $\epsilon_v$ that describes the trend during which the codimension 1 area teams incident to $\mathcal{G}_v$ are crossed by means of different facet teams incident to $\mathcal{G}_v$, and the crossing graph situation calls for that $\epsilon_v$ be attached or empty

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Additional info for Quasi-actions on trees II: Finite depth Bass-Serre trees

Example text

By connectivity of v , there exists an edge space Xe incident to Xv that crosses c Xe = Z in Xv , and so Xe also crosses Z in X, contradicting Step 1. 42 LEE MOSHER, MICHAH SAGEEV, AND KEVIN WHYTE Case 2: Suppose that for each Xe ∈ Verts( v ), Xe has deep intersection with exactly one of U − , U + , and so Xe may be labelled either positive or negative, depending on whether Xe has deep intersection with U + or with U − . Since the action of Stab(Xv ) on Xv is cocompact, the set U + contains an R-deep point that lies on some Xe ∈ v .

By the connect-thedots principle, it follows that each of the maps φev can be moved a bounded distance to obtain a continuous, cellular map Φev , so that the maps Φev have a common properness gauge, and so that the restrictions of Φev to the cells of Ye have only ﬁnitely many diﬀerent topological types up to pre and postcomposition. Therefore, by using the maps Φev to glue together the spaces ( v∈T Yv ) ∪ ( e⊂T (Ye × e)), we obtain a tree of spaces Y with the structure of a bounded geometry, uniformly contractible cell complex, and the maps ha piece together to give a quasi-isometry 26 LEE MOSHER, MICHAH SAGEEV, AND KEVIN WHYTE h : X → Y .

It follows that the restriction homomorphism Hcn−1 (NRi+k A) → Hcn−1 (NRi−1 A) factors through the restriction homomorphism Hcn−1 (NSi+k B) → Hcn−1 (NSi B). It therefore suﬃces to prove that for each S > 0 there exists S > S such that the restriction homomorphism Hcn−1 (NS B) → Hcn−1 (NS B) is zero, and so that the diﬀerence S − S depends only on Z. By applying Coarse Alexander Duality and a diagram chase, it suﬃces to prove that H0 (Z, Z − NS B) is zero for all S. But this is immediate because Z is connected and NS B Z.