Group presentations corresponding to spines of 3-manifolds. III
Group presentations corresponding to spines of 3-manifolds. III
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对应于 3 流形脊柱的组演示。
DOI：
10.1090/s0002-9947-1977-0488062-9
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发表时间：
1977
影响因子：
1.3
通讯作者：
R. S. Stevens
中科院分区：
文献类型：
--
作者：
R. Osborne;R. S. Stevens
在本系列的前几篇文章之后，我们将继续关注具有两个城镇的rr系统（即具有与具有两个生成器的群表示相对应的脊的紧致3-流形）。注意到一种有趣的对称性，然后用它来推导一些有用的结果。具体来说，证明了以下定理：定理1。设0是与紧致可定向3流形的一个脊相对应的群表示，设w是O的一个只涉及两个生成器a和b的关联子。如果w是循环约简的，那么(a) w可以“倒写”(即，如果w = amlibmliam2b2…a.kbk，则w是bnkf的循环共轭…b 2am2bnla m)，或(b) w属于a和b上自由群的换子子群。定理2。(宽松的翻译)。如果4是一个有两个生成器的群表示，如果对应的2复合体k是一个封闭可定向3流形的脊梁，那么Kp是一个封闭可定向3流形的脊梁，如果并且（除了两个小的情况）只有当4有两个关联，并且在六种允许的音节类型中（每个生成器中有3种），恰好有四种出现奇数次。此外，两个关联词中的每一个都可以“倒写”。在[3]中，引入了rr系统，并证明了rr系统包含了关于群表示与可定向3流形脊之间关系的许多有用信息。本文主要研究具有两个城镇的rr系统，以及相应的具有两个发电机的群表示。我们观察到许多rr系统具有某种对称性。利用这种对称性，我们将得到一些有用的结果。特别地，我们将建立以下内容：定理1。设T为紧致可定向3流形的一个脊对应的群表示，设w为p的一个只涉及两个发生器a和b的相对子。如果w循环约简，则w可以“倒写”或位于a和b上自由群的换易子群中(具体地说，如果w = amIbn1amV12…Amkbnk是w = b的循环共轭。B 2a m2b n'am。(Cf[5])。编辑于1976年4月20日收到。AMS （MOS）学科分类（1970年）。主要57 a10;二次55 25。
Continuing after the previous papers of this series, attention is devoted to RR-systems having two towns (i.e., to compact 3-manifolds with spines corresponding to group presentations having two generators). An interesting kind of symmetry is noted and then used to derive some useful results. Specifically, the following theorems are proved: THEOREM 1. Let 0 be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of O involving just two generators a and b. If w is cyclically reduced, then either (a) w can be "written backwards" (i.e., if w = amlibmliam2b2 ... a.kbk, then w is a cyclic conjugate of bnkamf ... b 2am2bnla m), or (b) w lies in the commutator subgroup of the free group on a and b. THEOREM 2. (Loose translation). If 4 is a group presentation with two generators and if the corresponding 2-complex K. is a spine of a closed orientable 3-manifold then,Kp is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if 4 has two relators and among the six allowable types of syllables (3 in each generator), exactly four occur an odd number of times. Further, each of the two relators can be "written backwards." In [3] RR-systems were introduced and were shown to contain much useful information about the relationship between group presentations and spines of orientable 3-manifolds. In this paper we devote our attention to RR-systems having two towns and correspondingly to group presentations with two generators. We observe that many of these RR-systems have a certain kind of symmetry. By means of this symmetry we will derive some useful results. In particular, we will establish the following: THEOREM 1. Let T be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of p involving just two generators a and b. If w is cyclically reduced, then w can be "written backwards" or lies in the commutator subgroup of the free group on a and b. (Specifically, w can be "written backwards" if w = amIbn1amV12 ... amkbnk is a cyclic conjugate of w = b nka mk . . . b 2a m2b n'am.) (Cf. [5].) Received by the editors April 20, 1976. AMS (MOS) subject classifications (1970). Primary 57A10; Secondary 55A25.