Coarse medians as a hyperbolicity surrogate

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。在这种情况下，一个自然且特别具有挑战性的问题是理解有限生成群G的性质如何反映到它的外部自同构群Out(G)的性质上。已经确定了G会导致一些美丽而复杂的理论，比如映射类群和Out（F_n）的理论。但更值得注意的是，任意“负弯曲”群的自同构——负弯曲奇异空间的基本群——可以在完全一般的情况下进行研究，并显示出与r树和合并积分裂上的作用密切相关的丰富结构。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意义，因为许多几何起源的群只满足这种较弱的假设。不幸的是，没有一种经典的技巧延续到这个场景中，而且几乎没有迹象表明，在这种情况下，这幅画不应该完全狂野。然而，最近的证据表明，实际上有大量的结构调节“紧曲”群的自同构——分解成立方体的非正弯曲空间的基本群。这样的群仍然是非常普遍的，包括所有已知的具有有趣自同构的非正弯曲群。完整的图像仍有待揭示，它有潜力成为一种真正统一的方法，极大地推广了直角Artin群的自同构的最新突破，同时也进一步将它们与经典的Out（F_n）理论和映射类群联系起来。我提出的研究将在这个令人兴奋的项目中迈出第一步。首先，我将在这种极端的一般性水平上研究Out(G)，旨在证明它对所有紧育群G是有限生成的。一个重要的工具将是Out(G)对G的紧育空间及其自然分类的作用：粗中位数结构空间。其次，我将研究在稍微受限的特殊群（在Haglund-Wise意义上）和直角Artin群的自同构的一些更精细的问题。我将展示固定子群体的行为和增长率如何遵循一种既受限制又令人惊讶地丰富和多样化的模式。第三，我将通过开发针对非双曲群的新计算程序来证明紧计算群的类别甚至比预期的更广泛。在整个项目中，基本的新想法是通过“粗中间结构”来弥补负曲率的不足，这是Brian Bowditch最近引入的一个概念。

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.