Surface subgroups from linear programming
Surface subgroups from linear programming
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线性规划的曲面子群
DOI：
10.1215/00127094-2877511
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发表时间：
2012
影响因子：
2.5
通讯作者：
Alden Walker
中科院分区：
文献类型：
--
作者：
Danny Calegari;Alden Walker
我们证明了一个自由群的随机自同态产生一个HNN扩展，该扩展包含一个闭面子群，其概率为1；一个特殊的例子是与一个2秩自由群（Sapir认为）的自同态相关的HNN扩展，将a发送到ab，将b发送到ba。进一步证明了将自由群沿任意秩子群的集合加倍得到的群具有这样的性质，即每一个有理二维同调类都虚表示为面子群，并且Gromov范数中的单位球是有限边有理多面体。这些结果是通过组合规划、几何规划和线性规划技术的混合得到的。我们得到了关于自由群中稳定换向子长度的其他更技术性的相关结果，特别是关于所谓的计数拟同态空间的结构
We show that a random endomorphism of a free group gives rise to an HNN extension which contains a closed surface subgroup, with probability one; a special case is the HNN extension associated to the endomorphism of a rank 2 free group (considered by Sapir) sending a to ab and b to ba. We further show that a group obtained by doubling a free group along any collection of subgroups (of arbitrary rank) has the property that every rational 2-dimensional homology class is virtually represented by surface subgroups, and the unit ball in the Gromov norm is a finite sided rational polyhedron. These results are obtained by a mixture of combinatorial, geometric and linear programming techniques. We obtain other related results of a more technical nature concerning stable commutator length in free groups, especially concerning the structure of the space of so-called counting quasimorphisms