单端群上点灯器的渐近几何

IF 2.6 1区 数学 Q1 MATHEMATICS Inventiones mathematicae Pub Date : 2024-07-22 DOI:10.1007/s00222-024-01278-w
Anthony Genevois, Romain Tessera
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引用次数: 0

摘要

这篇文章致力于研究花环积的渐近几何:\(F\wr H := \left ( \bigoplus _{H} F \right ) \rtimes H\) 其中\(F\)是有限群,\(H\)是有限生成群。我们的第一个主要结果指出,从有限呈现的单端群到 \(F\wr H\) 的粗糙映射必须与 \(H\) 的左余集保持有界距离。我们的第二个主要结果是在后一个结果的基础上,对有限呈现的单端群上的两个点灯群之间的准等距进行了非常严格的描述。第三,我们得到了这些群的完整分类,直至准等轴性。更准确地说,给定两个有限群 \(F_{1}\),\(F_{2}\)和两个有限呈现的一端群 \(H_{1}\),\(H_{2}\)、我们证明当且仅当 (i) \(H_{1}\), \(H_{2}\) 是非可门的准等距群并且 \(|F_{1}|\)、\(|F_{2}|\)有相同的素除数,或者 (ii) \(H_{1}\),\(H_{2}\)是可相容的,\(|F_{1}|=k^{n_{1}}\)和\(|F_{2}|=k^{n_{2}}\)对于某个\(k、n_{1},n_{2} \geq 1\), 并且存在一个准((n_{2}/n_{1})\)-to-one 准等分线 \(H_{1} \to H_{2}\).这可以看作是埃斯金-费舍尔-怀特(Eskin-Fisher-Whyte)著名工作的深远扩展,他处理的是\(H=\mathbb{Z}\)的情况。然而,我们的方法有着本质的不同,因为它关键地利用了 \(H\) 是单端的假设。我们的核心工具是对涉及准中值空间自然族的点灯组的一种新的几何解释。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Asymptotic geometry of lamplighters over one-ended groups

This article is dedicated to the asymptotic geometry of wreath products \(F\wr H := \left ( \bigoplus _{H} F \right ) \rtimes H\) where \(F\) is a finite group and \(H\) is a finitely generated group. Our first main result says that a coarse map from a finitely presented one-ended group to \(F\wr H\) must land at bounded distance from a left coset of \(H\). Our second main result, building on the later, is a very restrictive description of quasi-isometries between two lamplighter groups on finitely presented one-ended groups. Third, we obtain a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups \(F_{1}\), \(F_{2}\) and two finitely presented one-ended groups \(H_{1}\), \(H_{2}\), we show that \(F_{1} \wr H_{1}\) and \(F_{2} \wr H_{2}\) are quasi-isometric if and only if either (i) \(H_{1}\), \(H_{2}\) are non-amenable quasi-isometric groups and \(|F_{1}|\), \(|F_{2}|\) have the same prime divisors, or (ii) \(H_{1}\), \(H_{2}\) are amenable, \(|F_{1}|=k^{n_{1}}\) and \(|F_{2}|=k^{n_{2}}\) for some \(k,n_{1},n_{2} \geq 1\), and there exists a quasi-\((n_{2}/n_{1})\)-to-one quasi-isometry \(H_{1} \to H_{2}\). This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of \(H=\mathbb{Z}\). Our approach is however fundamentally different, as it crucially exploits the assumption that \(H\) is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces.

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来源期刊
Inventiones mathematicae
Inventiones mathematicae 数学-数学
CiteScore
5.60
自引率
3.20%
发文量
76
审稿时长
12 months
期刊介绍: This journal is published at frequent intervals to bring out new contributions to mathematics. It is a policy of the journal to publish papers within four months of acceptance. Once a paper is accepted it goes immediately into production and no changes can be made by the author(s).
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