Convergence groups are Fuchsian groups-猫眼课题宝

收敛群是 Fuchsian 群

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1991

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David Gabai

证明了满足“收敛性”的圆的一组同胚是单位圆盘的离散Mobius变换组的限制。这完成了塞弗特纤维空间猜想的证明，并给出了Nielson实现问题的一个新证明。Fuchsian群F是R中单位圆盘D上Mobius变换群的离散子群。F限制于Homeo（5 ′）的子群G，它满足下列收敛性质[GM]。给定G的不同元素序列，则存在x，yeS和子序列{f.}在S {x，y} f上，-· y，f~ -> x在紧集上一致。具有这种性质的群G c Homeo（5 '）称为收敛群。我们宣布以下结果。细节可以在[G]中找到。定理1. G是收敛群当且仅当G在Homeo（5）中共轭到Fuchsian群的限制。一个Seifert流形空间是一个紧的3-流形M，它几乎是一个紧曲面上的S丛，即存在一个投影n：M -· TV使得对于每个x ∈ N，存在x的D邻域使得n~（D）= D × S和π t（（r，6X）9（1，02））=其中p ^ O和p，q是互质的并且取决于x和6 e Rmod 27 t。推论2（塞弗特纤维空间猜想）。设M是一个紧致的、可定向的、不可约的（即每个光滑嵌入的S都包围一个3-胞腔）3-流形，且n_x为无穷大，则M是一个Seifert连通空间当且仅当n_x（M）包含一个循环正规子群.编辑于1991年1月7日收到，修订版于1991年4月20日收到。1980年数学学科分类（1985年修订）。小学57 S25;中学20 H10、57 N 05、57 N10。部分由NSF Grant DMS-8902343和Sloan Foundation Fellowship支持。[2]安德鲁·卡森（Andrew Casson）也用不同的方法证明了定理1。©1991美国数学学会0273-0979/91 $1.00+ $.25每页395

A group of homeomorphisms of the circle satisfying the "convergence property" is shown to be the restriction of a discrete group of Mobius transformations of the unit disk. This completes the proof of the Seifert fiber space conjecture and gives a new proof of the Nielson realization problem. A Fuchsian group F is a discrete subgroup of the group of Mobius transformations on the unit disc D in R . F restricts to a subgroup G of Homeo(5') which satisfies the following convergence property [GM]. Given a sequence of distinct elements of G, then there exists x, y e S and a subsequence {f.} such that on S {x, y} f. -• y, f~ -> x uniformly on compact sets. A group G c Homeo(5') with this property is called a convergence group. We announce the following result. The details can be found in [G]. Theorem 1. G is a convergence group if and only if G is conjugate in Homeo(5) to the restriction of a Fuchsian group.* A Seifert fibred space is a compact 3-manifold M which is almost an S bundle over a compact surface, i.e. there exists a projection n : M —• TV such that for each x e N there exists a D neighborhood of x such that n~(D) = D x S and 7t((r, 6X)9 (1, 02)) = (r9pOx + Q62) where p ^ O and p, q are relatively prime and depend on x and 6 e Rmod27t. Corollary 2 (Seifert Fibred Space Conjecture). Let M be a compact, orientable, irreducible {i.e. every smooth embedded S bounds a 3-cell) 3-manifold with infinite nx, then M is a Seifert fibred space if and only if nx(M) contains a cyclic normal subgroup. Received by the editors January 7, 1991 and, in revised form, April 20, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 57S25; Secondary 20H10, 57N05, 57N10. Partially supported by NSF Grant DMS-8902343 and a Sloan Foundation Fellowship. *Andrew Casson has also announced, using different methods, a proof of Theorem 1. ©1991 American Mathematical Society 0273-0979/91 $1.00+ $.25 per page 395

Convergence groups are Fuchsian groups

收敛群是 Fuchsian 群

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