Bounded Scale Measure and Property A

不具有有界指数性质的非球面流形

Aspherical manifolds which do not have Bounded Index Property

本文证明某些非球面流形不具有有界指数性质，从而否定了Jiang提出的相关问题。

On the bounded index property for products of aspherical polyhedra

A compact polyhedron $X$ is said to have the Bounded Index Property for Homotopy Equivalence (BIPHE) if there is a finite bound $\mathcal{B}$ such that for any homotopy equivalence $f:X\rightarrow X$ and any fixed point class $\mathbf{F}$ of $f$, the index $|\mathrm{ind}(f,\mathbf{F})|\leq \mathcal{B}$. In this note, we consider the product of compact polyhedra, and give some sufficient conditions for it to have BIPHE. Moreover, we show that the products of closed Riemannian manifolds with negative sectional curvature, in particular hyperbolic manifolds, have BIPHE, which gives an affirmative …

Linearly Bounded Conjugator Property for Mapping Class Groups

Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.

关于 Jiang 有界指数性质在幂零流形乘积上的研究

On Jiang's Bounded Index Property for products of nilmanifolds

本文推广有界指数性质至迭代情形，证明特定幂零流形乘积满足该性质，并刻画其自同伦等价的不动点相关数。

Central extensions and bounded cohomology

It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a…

Packing subgroups in solvable groups

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated subgroup without the bounded packing property. In this example the subgroup is a metabelian retract also. Thus we obtain a negative answer to Problem 2.27 of Hruska-Wise. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded pack…