论文标题
移动空间和希格曼 - 汤普森组的自动形态：双面情况
Automorphisms of shift spaces and the Higman-Thompson groups: the two-sided case
论文作者
论文摘要
在本文中，我们进一步探讨了全班空间的自动形态组与希格曼的外部自动形态组之间的联系 - 汤普森组$ \ {g_ {g_ {n，r} \} $。 我们表明，（双面）移动动态系统的自动形态的商$ \ mathrm {aut}（x_n^{\ mathbb {n}}，σ_{n}），其中心嵌入其特定的子集团$ \ mathcal $ \ mathcal {l} _ {l} _ { $ g_ {n，n，n-1} $的$ \ mathop {\ mathrm {out}}} $ of $ g_ {n，n-1} $。以下是瑞安（Ryan）的结果，我们具有以下中心扩展：$ \langleσ_{n} \ rangle \ hookrightArrow \ hookrightArrow \ mathrm {aut}（x_n^{\ mathbb {n}} $ \langleσ_{n} \ rangle \ cong \ mathbb {z} $。我们证明，当且仅当$ n $不是适当的功率时，此简短的序列会拆分，并且在所有情况下，我们为扩展程序计算了2个循环和2个循环系统。我们还使用此中央扩展名来证明，对于$ 1 \ le r <n $，$ \ mathop {\ mathrm {out}}}（g_ {n，r}）$是无中心的，并且存在不可确定的订单问题。 请注意，组$ \ mathop {\ mathrm {out}}（g_ {n，n，n-1}）$由有限的换能器（在自动机理论中产生的组合对象）组成，并且内部$ \ nathcal {l} _ {n} $的元素很容易表征$ \ mathop {\ mathrm {out}}（g_ {n，n，n-1}）$通过简单的组合属性。特别是，简短的序列使我们能够确定$ \ mathrm {aut}元素的新的，有效的纯粹组合表示（x_n^{\ mathbb {n}}，σ_{n}）$，我们演示了如何使用此新表示来计算产品。
In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman--Thompson groups $\{G_{n,r}\}$. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$ by its centre embeds as a particular subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$\langle σ_{n}\rangle \hookrightarrow \mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n}) \twoheadrightarrow \mathcal{L}_{n}$$ where here, $\langle σ_{n} \rangle \cong \mathbb{Z}$. We prove that this short exact sequence splits if and only if $n$ is not a proper power, and, in all cases, we compute the 2-cocycles and 2-coboundaries for the extension. We also use this central extension to prove that for $1 \le r < n$, the groups $\mathop{\mathrm{Out}}(G_{n,r})$ are centreless and have undecidable order problem. Note that the group $\mathop{\mathrm{Out}}(G_{n,n-1})$ consists of finite transducers (combinatorial objects arising in automata theory), and elements of the group $\mathcal{L}_{n}$ are easily characterised within $\mathop{\mathrm{Out}}(G_{n,n-1})$ by a simple combinatorial property. In particular, the short exact sequence allows us to determine a new and efficient purely combinatorial representation of elements of $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$, and we demonstrate how to compute products using this new representation.
