arXiv 1708.045912017-08math.GRmath.LO

缺双曲群的字与共轭问题

The word and conjugacy problems in lacunary hyperbolic groups

Arman Darbinyan

备注：: updated Definition 2.2, minor fixes

摘要：

研究缺双曲群（简称LHG）中的字与共轭问题。特别地，我们描述了LHG中字问题可判定的一个充要条件。然后，基于Olshanskii的分次小消去理论，我们发展了一个一般框架，使我们能够构造缺双曲群的字和共轭问题高度可控和灵活的可计算性和计算复杂性。作为一个应用程序，我们表明，对于任何递归可重复的子集$\mathcal{L} \subseteq \mathcal{A}^*$，其中$\mathcal{A}^*$是一组字在任意选择的非空有限字母表$\mathcal{A}$，存在一个缺双曲群$G_{\mathcal{L}}$，使$ \mathcal{L}$的成员资格问题是“几乎”线性时间等价于共轭问题$G_{\mathcal{L}}$。此外，对于上述组的字和个人的共轭问题是可判定的“几乎”线性时间。另一个应用是建设一个缺双曲群与“几乎”线性时间字问题和所有的个人共轭问题是不可判定的字问题。作为另一个应用程序的开发框架，我们构建无限的口头完全群和扭转免费塔斯基怪物，即无限扭转免费组的所有真子群是循环的，“几乎”线性时间字和多项式时间共轭问题。这些群被构造为任意给定的非初等无挠双曲群的代数，并且是缺双曲群。最后，作为主要结果的结果，我们回答了一些开放的问题。

英文摘要：

We study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity. As an application, we show that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathcal{A}^*$ is the set of words over arbitrarily chosen non-empty finite alphabet $\mathcal{A}$, there exists a lacunary hyperbolic group $G_{\mathcal{L}}$ such that the membership problem for $ \mathcal{L}$ is `almost' linear time equivalent to the conjugacy problem in $G_{\mathcal{L}}$. Moreover, for the mentioned group the word and individual conjugacy problems are decidable in `almost' linear time. Another application is the construction of a lacunary hyperbolic group with `almost' linear time word problem and with all the individual conjugacy problems being undecidable except the word problem. As yet another application of the developed framework, we construct infinite verbally complete groups and torsion free Tarski monsters, i.e. infinite torsion-free groups all of whose proper subgroups are cyclic, with `almost' linear time word and polynomial time conjugacy problems. These groups are constructed as quotients of arbitrarily given non-elementary torsion-free hyperbolic groups and are lacunary hyperbolic. Finally, as a consequence of the main results, we answer a few open questions.

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摘要：

英文摘要：

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