Manifolds counting and class field towers

Rank gradient in co-final towers of certain Kleinian groups

We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedra in $\mathbb{H}^3$ then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co-final towers of covers with zero rank gradient.

双曲三维流形中浸入曲面的计数

Counting immersed surfaces in hyperbolic 3-manifolds

本文计算了双曲三维流形群中极大亏格g曲面子群的共轭类的个数。对任意闭双曲3-流形，我们证明了这个数存在一个上界，它随g的因子增长。我们还给出了一类闭双曲3-流形，它存在一个同类型的下界。

Counting cusped hyperbolic 3-manifolds that bound geometrically

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.

直到切同伦等价的开负曲流形的计数

Counting open negatively curved manifolds up to tangential homotopy equivalence

在群G的适当假设下，证明了具有一致有界负截面曲率且基本群同构于群G的完备黎曼n-流形类分裂成100多个切同伦类型.因此，许多非球面流形不允许具有规定曲率界的完全负曲度量。

Counting isospectral manifolds

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [BL]. Our proof uses Sunada's method, results of [BL], and some deep results from number theory. We also discuss an open number-theoretical problem w…

计数双陪集及其在一般三维流形中的应用

Counting double cosets with application to generic 3-manifolds

本研究推广了双陪集增长理论至更广群类，证明 Morse 子群双陪集增长与轨道增长相当，并应用于证实 Maher 猜想：双曲三维流形在 Heegaard 分解流形中指数泛型。