Rank gradient in co-final towers of certain Kleinian groups

Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

临界指数小于1的可及CAT(-1)群

Accessible CAT$(-1)$ groups of critical exponent less than one

该研究针对可及CAT(-1)群，证明临界指数小于1时群的结构性质，给出相关推论，覆盖更大类群并补充JSJ相关结论。

Knot concordance, Whitney towers and L^2 signatures

We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain {\em universal} groups. Finally, we…

Semifree circle actions, Bott towers, and quasitoric manifolds

A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This …

通过弱一般 Nakano 消没定理研究 L²-Betti 数与规范化 Hodge 数的收敛性

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

本文利用弱一般 Nakano 消没定理，研究 Abel 覆盖塔中规范化 Hodge 数与多重亏格的增长及收敛，并计算相关非正则簇的 L²-Betti 数，推广 Kollár 的可乘性结果。

Free subgroups of $3$-manifold groups

We show that any closed hyperbolic $3$-manifold $M$ has a co-final tower of covers $M_i \to M$ of degrees $n_i$ such that any subgroup of $π_1(M_i)$ generated by $k_i$ elements is free, where $k_i \ge n_i^C$ and $C = C(M) > 0$. Together with this result we show that $\log k_i \geq C_1 sys_1(M_i)$, where $sys_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic $3$-manifolds.