$M_g$ 的子集的粗密度

超限分离与超限分解在双曲群中的研究

Coarse separation and splittings in hyperbolic groups

本文从定量角度研究双曲群的粗分离，证明了非虚拟曲面群的双曲群在满足特定条件时可被子集粗分离，且给出了双曲群的分离轮廓下界及图积的粗分离条件。

Asymptotic dimension and small subsets in locally compact topological groups

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean space $R^n$. Also we prove that for a locally compact Abelian group $X$, the equality $S(X)=D_<(X)$ holds if and only if the group $X$ is compactly generated.

Generic coarse geometry of leaves

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely qu…

Convexity in hierarchically hyperbolic spaces

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting properties, relative divergence, the coarse median structure, and the hierarchical structure itself. Along the way, we obtain new tools to study HHSs, including two new equivalent definitions of hierarchical quasiconvexity and a version of the bounded geodesic image property for str…

粗负曲率的测地流形

Geodesic currents of coarse negative curvature

本文研究了粗负曲率下的测地流形，证明了其双伪度量为强粗负曲率的测地流形在测地流形空间中稠密，并展示了非正曲率流形的测地流形不稠密，进而构造了非CAT(0)的强粗负曲率测地度量。

A new construction of universal spaces for asymptotic dimension

For each $n$, we construct a separable metric space $\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\mathop{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\mathop{udim}$) at most $n$. Thus, $\mathbb{U}_n$ serves as a universal space for dimension $n$ in both the large-scale and infinitesimal topology. More precisely, we prove: \[ \mathop{asdim} \mathbb{U}_n = \mathop{udim} \mathbb{U}_n = n \] and such that for each separable metric space $X$, a) if $\mathop{asdim} X \l…