Widths of subgroups.

*(English)*Zbl 0897.20030Let \(H\) 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.

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.

Reviewer: D.McCullough (Norman)

##### Keywords:

essentially distinct elements; widths of subgroups; malnormal subgroups; \(k\)-plane property; quasiconvex subgroups; negatively curved groups; visual metrics; limit sets; essentially distinct conjugates of subgroups; heights; fundamental groups; Fuchsian groups
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\textit{R. Gitik} et al., Trans. Am. Math. Soc. 350, No. 1, 321--329 (1998; Zbl 0897.20030)

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##### References:

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