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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 Geom. 35, 103-130 (1992)] and A. Casson (unpublished) about homeomorphisms of the universal covers of such 3-manifolds to \(\mathbb{R}^ 3\).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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[1] R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365 – 383. · Zbl 0148.37202
[2] Matthew G. Brin and T. L. Thickstun, Open, irreducible 3-manifolds which are end 1-movable, Topology 26 (1987), no. 2, 211 – 233. · Zbl 0622.57010 · doi:10.1016/0040-9383(87)90062-0 · doi.org
[3] Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. · Zbl 0153.52905
[4] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. · Zbl 0584.20036
[5] A. Casson, lecture at Stanford Univ., March 1990.
[6] T. A. Chapman, Lectures on Hilbert cube manifolds, American Mathematical Society, Providence, R. I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975; Regional Conference Series in Mathematics, No. 28. · Zbl 0317.57009
[7] Steve Ferry, Homotoping \?-maps to homeomorphisms, Amer. J. Math. 101 (1979), no. 3, 567 – 582. , https://doi.org/10.2307/2373798 T. A. Chapman and Steve Ferry, Approximating homotopy equivalences by homeomorphisms, Amer. J. Math. 101 (1979), no. 3, 583 – 607. · Zbl 0426.57004 · doi:10.2307/2373799 · doi.org
[8] T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976), no. 3-4, 171 – 208. · Zbl 0361.57008 · doi:10.1007/BF02392417 · doi.org
[9] Michael W. Davis and Tadeusz Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991), no. 2, 347 – 388. · Zbl 0723.57017
[10] Jerzy Dydak and Jack Segal, Shape theory, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. An introduction. · Zbl 0401.54028
[11] A. N. Dranishnikov, On a problem of P. S. Aleksandrov, Mat. Sb. (N.S.) 135(177) (1988), no. 4, 551 – 557, 560 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 539 – 545. · Zbl 0643.55001
[12] Steve Ferry, Homotoping \?-maps to homeomorphisms, Amer. J. Math. 101 (1979), no. 3, 567 – 582. , https://doi.org/10.2307/2373798 T. A. Chapman and Steve Ferry, Approximating homotopy equivalences by homeomorphisms, Amer. J. Math. 101 (1979), no. 3, 583 – 607. · Zbl 0426.57004 · doi:10.2307/2373799 · doi.org
[13] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. · Zbl 0727.20018
[14] S. M. Gersten , Essays in group theory, Mathematical Sciences Research Institute Publications, vol. 8, Springer-Verlag, New York, 1987. · Zbl 0626.00014
[15] David W. Henderson, \?-sets in ANR’s, Trans. Amer. Math. Soc. 213 (1975), 205 – 216. · Zbl 0315.57004
[16] W. Jakobsche, Approximating homotopy equivalences of surfaces by homeomorphisms, Fund. Math. 118 (1983), no. 1, 1 – 9. · Zbl 0555.57002
[17] V. Poenaru, Almost convex groups, Lipschitz combing, and \( \pi _1^\infty \) for universal covering spaces of closed \( 3\)-manifolds, Orsay prépublication 90-09.
[18] E. Ghys and P. de la Harpe , Sur les groupes hyperboliques d’aprés Mikhael Gromov, Birkhäuser, 1990. · Zbl 0731.20025
[19] T. Brady, D. Cooper, T. Delzant, M. Lustig, M. Mihalik, M. Shapiro, H. Short, and A. N. Other, Notes on negatively curved groups.
[20] John J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 105 – 118. · Zbl 0474.55002
[21] Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, vol. 32, American Mathematical Society, New York, N. Y., 1949. · Zbl 0039.39602
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