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Widths of subgroups. (English) Zbl 0897.20030
Let \(H\) be a subgroup of \(G\). Elements \(g_i\), \(1\leq i\leq n\), of \(G\) are called essentially distinct if \(Hg_i\neq Hg_j\) for \(i\neq j\). Conjugates of \(H\) by essentially distinct elements are called essentially distinct conjugates (although they may be the same subgroup of \(G\)). The authors define the width of \(H\) to be \(n\) if there exists a collection of \(n\) essentially distinct conjugates of \(H\) such that the intersection of all the elements of the collection is infinite and \(n\) is maximal (if \(H\) is finite, its width is \(0\)). The width is an algebraic version of the \(k\)-plane property introduced by P. Scott [Ann. Math., II. Ser. 117, 35-70 (1983; Zbl 0516.57006)]. Roughly speaking, it says that out of any \(k\) distinct planes corresponding to lifts of an immersed surface in a 3-manifold to the universal cover, at least two are disjoint. J. Hass and P. Scott [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)] used the 4-plane property to prove that homotopy equivalence implies homeomorphism for certain 3-manifolds.
The main result of the paper is that a quasiconvex subgroup of a negatively curved group has finite width. This implies that any quasi-Fuchsian subgroup of the fundamental group of a closed hyperbolic 3-manifold has the \(k\)-plane property for some \(k\). The authors give two proofs of the main theorem from different perspectives. The first and third authors use the basic geometry of negatively curved groups. The other two authors use the limit sets of negatively curved groups. Among the steps in their argument is the fact that if \(H_1,\ldots,H_n\) are quasiconvex subgroups of \(G\), then the intersection of their limit sets is the limit set of their intersection; this generalizes a result of P. Susskind and G. A. Swarup [Am. J. Math. 114, No. 2, 233-250 (1992; Zbl 0791.30039)] for the case when \(G\) is the fundamental group of a closed hyperbolic manifold.

MSC:
20F65 Geometric group theory
57M07 Topological methods in group theory
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[1] 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
[2] É. Ghys and P. de la Harpe , Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. · Zbl 0731.20025
[3] R. Gitik and E. Rips, Heights of Subgroups, MSRI Preprint 027-95. · Zbl 0838.20026
[4] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3 · doi.org
[5] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1 – 295. · Zbl 0841.20039
[6] Joel Hass and Peter Scott, Homotopy equivalence and homeomorphism of 3-manifolds, Topology 31 (1992), no. 3, 493 – 517. · Zbl 0771.57007 · doi:10.1016/0040-9383(92)90046-K · doi.org
[7] I. Kapovich and H. Short, Some Remarks on Quasiconvexity, preprint.
[8] M. Mitra, Immersed Incompressible Surfaces in Hyperbolic \(3\)-Manifolds, in preparation.
[9] H. Rubinstein and M. Sageev, Intersection Patterns of Immersed Incompressible Surfaces, in preparation. · Zbl 0924.57023
[10] Peter Scott, There are no fake Seifert fibre spaces with infinite \?\(_{1}\), Ann. of Math. (2) 117 (1983), no. 1, 35 – 70. · Zbl 0516.57006 · doi:10.2307/2006970 · doi.org
[11] Hamish Short, Quasiconvexity and a theorem of Howson’s, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 168 – 176. · Zbl 0869.20023
[12] Perry Susskind and Gadde A. Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992), no. 2, 233 – 250. · Zbl 0791.30039 · doi:10.2307/2374703 · doi.org
[13] E. Swenson, Limit Sets in the Boundary of Negatively Curved Groups, preprint.
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