Coarse medians as a hyperbolicity surrogate

粗中位数作为双曲性替代项

基本信息

项目摘要

Geometric group theory is the study of infinite, finitely generated groups through their actions on metric spaces. It is a new and dynamic field, thriving on its many connections to algebra, low-dimensional topology and differential geometry. A natural and particularly challenging problem in this setting has been understanding how the properties of a finitely generated group G will reflect onto those of its outer automorphism group Out(G). Already fixing G leads to beautiful and complicated theories, such as those of mapping class groups and Out(F_n). But what is even more remarkable is that automorphisms of arbitrary “negatively curved'' groups -- fundamental groups of negatively curved singular spaces -- can be studied in full generality, and display a rich structure that is intimately related to actions on R-trees and amalgamated-product splittings. Gaining a similar understanding of Out(G) when G is just “non-positively curved'' would have far-reaching implications, as many groups of a geometric origin only satisfy this weaker assumption. Unfortunately, none of the classical techniques carry over into this setting, and there is little indication that the picture should not be completely wild in this generality. Recent evidence suggests, however, that there is actually a great deal of structure regulating automorphisms of “cocompactly cubulated'' groups -- fundamental groups of non-positively curved spaces with a decomposition into cubes. Such groups are still very general and include all known non-positively curved groups with interesting automorphisms. The full picture is still waiting to be uncovered here and it has the potential to become a truly unifying approach, greatly generalising recent breakthroughs on automorphisms of right-angled Artin groups, while also further connecting them to the classical theory of Out(F_n) and mapping class groups. My proposed research will take the first steps in this exciting program. First, I will study Out(G) in this extreme level of generality, aiming to show that it is finitely generated for all cocompactly cubulated groups G. An important tool will be the action of Out(G) on the space of cubulations of G and on its natural coarsification: the space of coarse median structures. Second, I will investigate some of the finer questions on automorphisms in the slightly more restricted setting of special groups (in the Haglund-Wise sense) and right-angled Artin groups. I will demonstrate how the behaviour of fixed subgroups and growth rates, in particular, follows a pattern that is both restrained and surprisingly rich and variegated. Third, I will show that the class of cocompactly cubulated groups is even broader than expected, by developing new cubulating procedures taylored to non-hyperbolic groups. The fundamental new idea, throughout the project, is to make up for the lack of negative curvature by means of a “coarse median structure'', a concept recently introduced by Brian Bowditch.
几何群论是研究无限的、有限生成的群在度量空间上的作用。它是一个新的、动态的领域,因其与代数、低维拓扑和微分几何的许多联系而蓬勃发展。在这一背景下,一个自然且特别具有挑战性的问题是理解有限生成群G的性质将如何映射到它的外自同构群OUT(G)的性质。已有的固定G产生了美丽而复杂的理论,如映射类群和Out(F_N)的理论。但更值得注意的是,任意“负曲”群--负曲奇异空间的基本群--的自同构可以被全面地研究,并显示出与R-树上的作用和合并积分裂密切相关的丰富的结构。当G只是“非正弯曲”时,获得对Out(G)的类似理解将具有深远的影响,因为许多几何起源的群只满足这一较弱的假设。不幸的是,没有一种经典的技术延续到这种设置中,而且几乎没有迹象表明,在这种普遍情况下,画面不应该完全狂野。然而,最近的证据表明,实际上存在大量的结构来调节“紧立方”群的自同构--非正曲线空间的基本群,并将其分解为立方体。这样的群仍然是非常一般的,并且包括所有已知的具有有趣的自同构的非正曲群。这一方法有可能成为一种真正统一的方法,极大地推广了直角Artin群自同构的最新突破,同时也进一步将它们与Out(F_N)和映射类群的经典理论联系起来。我提出的研究将在这个令人兴奋的计划中迈出第一步。首先,我将在这个极端的一般水平上研究(G),目的是证明它是对所有紧集合群G有限生成的。一个重要的工具将是Out(G)在G的立方空间及其自然粗化:粗中结构空间上的作用。其次,我将在稍微受限的特殊群(在Haglund-Wise意义下)和直角Artin群的背景下研究关于自同构的一些更精细的问题。我将展示固定分组和增长率的行为是如何遵循一种既有节制又令人惊讶地丰富和多样化的模式的。第三,我将通过开发非双曲群的泰洛德新的立群程序来证明紧立群的类比预期的更广泛。在整个项目中,基本的新想法是通过“粗略的中间结构”来弥补负曲率的不足,这是布莱恩·鲍迪奇最近提出的一个概念。

项目成果

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Dr. Elia Fioravanti其他文献

Dr. Elia Fioravanti的其他文献

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