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The topology at infinity of $$\text{Out} (F_n)$$. (English) Zbl 0954.55011
A locally compact space is said to be $$r$$-connected at infinity provided for every compact set $$K\subset X$$ there exists a compact set $$L\subset X$$ such that $$L\supset K$$ and any map $$\alpha :S^d\to X\smallsetminus L$$ with $$d\leq r$$ extends to a map $$\widetilde{\alpha}:D^{d+1}\to X\smallsetminus K$$. Similarly, a discrete group $$\Gamma$$ is said to be $$r$$-connected at infinity if it acts cocompactly, freely and simplicially on an $$r$$-connected simplicial complex $$X$$ which is $$r$$-connected at infinity. We note that if $$\Gamma$$ acts cocompactly, freely and simplicially on another $$r$$-connected simplicial complex $$Y$$, then $$Y$$ is necessarily also $$r$$-connected at infinity. There are several natural contractible spaces on which $$\text{Out}(F_n)$$ acts, namely Culler-Vogtmann’s Outer space and its variants. Outer space is to be regarded as the analog of homogeneous space with the action of an arithmetic group or the Teichmüller space with the action of the mapping class group of a compact surface.
In this paper the authors prove that the group $$\text{Out}(F_n)$$ is $$(2n-5)$$-connected at infinity if $$n\geq 2$$. To obtain this result, they construct the bordification of Outer space and develop a Morse theory that applies to (bordified) Outer space in the style introduced by the first author and N. Brady [Invent. Math. 129, No. 3, 445-470 (1997; Zbl 0888.20021)]. As a corollary of their analysis of the topology of bordified Outer space they also obtain that $$\text{Out}(F_n)$$ is a virtual duality group of dimension $$2n-3$$ if $$n\geq 2$$.

##### MSC:
 55P15 Classification of homotopy type 57M07 Topological methods in group theory 20F65 Geometric group theory
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