Controlled coarse homology and isoperimetric inequalities

Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for …

局部紧群的等周轮廓与正则嵌入

Isoperimetric Profiles and Regular Embeddings of locally compact groups

本文将 $L^p$-测度子群耦合推广至局部紧紧生成单模群，证明了顺从群间正则嵌入的动力学准则，并得出 $L^p$-等周轮廓在正则嵌入与粗嵌入下保持单调性的结论。

自由因子复形的组合等周不等式

Combinatorial isoperimetric inequality for the free factor complex

本研究证明秩≥4的自由群的自由因子复形不满足组合等周不等式，通过构造粗Lipschitz函数，说明存在长度为4的环需填充的2-单形数量随N线性增长。

Uniformly finite homology and amenable groups

Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and prove that it is infinite dimensional in many cases. The main idea is to use different transfer maps to distinguish between classes in uniformly finite homology. Furthermore we show that there are infinitely many classes in degree zero that cannot be detected by means.

对的粗亚历山大对偶及其应用

Coarse Alexander duality for pairs and applications

本文建立切赫同调的相对版本并证明粗亚历山大对偶的相对形式，进而利用 Jordan 环不变量证明存在一个实际 Kleinian 但本身非 Kleinian 的 3-流形群。

Dynamic characterizations of quasi-isometry, and applications to cohomology

We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property $H_{FD}$. As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type $FP_n$ (over arbitrary rings) is a coarse inv…