The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups
The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups
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表面群猜想：循环收缩和共轭收缩一关系群
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
G. Rosenberger
中科院分区：
文献类型：
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作者：
L. Ciobanu;B. Fine;G. Rosenberger
一般的曲面群猜想问一个单关系群是否是一个曲面群，其中每个有限指数的子群都是单关系群，并且每个无限指数的子群都是自由的（性质IF）。我们解决了几个相关的acquitures在罚款等。(Sci Math A 1：1-15，2008）。首先，我们得到了循环压缩和共轭压缩单关系子群的曲面群猜想B。那就是：若G是满足性质IF的循环压缩单关系群或共轭压缩单关系群，则G是自由的，是曲面群或可解的Baumslag-Solitar群. Fine et al. (Sci Math A 1：1-15，2008）在性质IF上，利用Wilton（Geom Topol，2012）的定理和Stallings（Ann Math 2（88）：312-334，1968）以及Kharlampovich和Myasnikov（Trans Am Math Soc 350（2）：571-613，1998）的结果，我们表明Fine等人提出的表面群猜想C。(Sci Math A 1：1-15，2008）为真，即：如果G是一个具有性质IF的非自由生成的自由不可分解的完全剩余自由群，则G是一个曲面群。
The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.