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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 Geom. 35, 103-130 (1992)] and A. Casson (unpublished) about homeomorphisms of the universal covers of such 3-manifolds to $$\mathbb{R}^ 3$$.}

##### MSC:
 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
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