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The boundary of negatively curved groups. (English) Zbl 0767.20014
Let \(G\) be a negatively curved group in the sense of M. Gromov [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] and \(P(G)=P_ d(G)\) be the Rips complex for \(G\) with \(d\) as sufficiently large as \(P(G)\) is contractible. Here a connection \(x_ 1,\ddots ,x_ k\in G\) spans a simplex in \(P_ d(G)\) if \(d(x_ i,x_ j)\leq d\) for all \(i,j\) where \(d(x,y)\) is the word metric on \(G\). Considering the boundary \(\partial G\) as the set of equivalence classes of sequences convergent at infinity, \(P(G)\) relates \(\partial G\) with the cohomological properties of \(G\).
The main observation of the paper is: Theorem 1.2: \(\widetilde{P(G)}=P(G)\cup \partial G\) is an absolute retract, and \(\partial G\subset \widetilde{P(G)}\) is a {\(bbfZ\)-set, i.e. for every open set \(U\subset \widetilde{P(G)}\) the inclusion \(U\partial G\hookrightarrow U\) is a homotopy equivalence.
As applications of this result, the authors discuss the local connectivity of \(\partial G\) and prove that the universal covers of closed, irreducible 3-manifolds with infinite negatively curved fundamental group G compactified by \(\partial G\) are homeomorphic to the 3-ball. This is a sharpening of the result of V. Poenaru [J. Differ. Geom. 35, 103-130 (1992)] and A. Casson (unpublished) about homeomorphisms of the universal covers of such 3-manifolds to \(\mathbb{R}^ 3\).}

20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20M05 Free semigroups, generators and relations, word problems
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
20J05 Homological methods in group theory
20F05 Generators, relations, and presentations of groups
Full Text: DOI
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