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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 Teichm├╝ller 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 \(n\geq 2\).

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