Permutation Groups with a Cyclic Two-Orbits Subgroup and Monodromy Groups of Siegel Functions

$\ell^{2}$-torsion of free-by-cyclic groups

We provide an upper bound on the $\ell^{2}$-torsion of a free-by-cyclic group, $-ρ^{(2)}(\mathbb{F} \rtimes_Φ \mathbb{Z})$, in terms of a relative train-track representative for $Φ\in \mathrm{Aut}(\mathbb{F})$. Our result shares features with a theorem of Lück-Schick computing the $\ell^{2}$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $\ell^{2}$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Lück-Schick, a special case of our bound is analogous to the bound on the volume of a 3-manifold th…

关于自由循环群和亏度1群的子群可分性

On subgroup separability of free-by-cyclic and deficiency 1 groups

本文证明了多项式增长单演的自由循环群在特定条件下子群可分，并指出随机亏度1群大概率不可分。

多项式增长的自由循环群的挠同调增长

Torsion homology growth of polynomially growing free-by-cyclic groups

本研究证明多项式增长单值群的自由循环群同调挠增长各维度均为零且与Farber链选取无关，推导出整挠等于ℓ²-挠，验证了Lück的相关猜想。

自由-循环群的最大酰柱形作用

Largest acylindrical actions of free-by-cyclic groups

该研究证明了自由-循环群存在最大的酰柱形作用，并刻画了其Morse测地线、稳定子群及Morse边界。

自由-循环群是共轭可分离的

Free-by-cyclic groups are conjugacy separable

本文证明了所有有限生成的自由-循环群都是共轭可分离的，并展示了外自同构群的残余有限性。

Dehn functions of subgroups of right-angled Artin groups

We show that for each positive integer $k$ there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as $n^k$. As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as $n^{k+2}$.

单端双曲群循环扩张中的共轭问题

The conjugacy problem in cyclic extensions of one-ended hyperbolic groups

本研究针对无挠单端双曲群的自同构映射环面，通过典范JSJ树悬垂、Nielsen–Thurston约化细化及分块折叠得到余紧柱形分裂，将共轭问题归约为有理集判定，证明了这类循环扩张群的共轭问题可解。

关于自由-循环群的 profinite 刚性 I：一般情况

On profinite rigidity amongst free-by-cyclic groups I: the generic case

本文研究了自由-循环群的 profinite 刚性，证明了在特定条件下，超几何性、纤维秩和特征多项式等是 profinite 完成的不变量，并展示了一般自由-循环群的几乎刚性性质。

两生成元单关系逆幺半群的字问题

The word problem for two-generator one-relator inverse monoids

本文研究两生成元单关系逆幺半群的字问题，证明了其可判定与不可判定结果，给出首个不可判定字问题的E-酉两生成元单关系逆幺半群等例子，还给出其可判定字问题的充分条件。

Small $C^1$ actions of semidirect products on compact manifolds

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $ψ$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $ψ^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $π_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provid…

Non-crossing partitions and Milnor fibers

For a finite real reflection group $W$ we use non-crossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$-discriminant $Δ_W$ and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of $Δ_W$.

性质(LR)与虚自由群的嵌入定理

Property (LR) and an embedding theorem for virtually free groups

本文证明了每个虚自由群G具有Long和Reid提出的性质(LR)，即每个有限生成子群都是有限指数子群的 retract。主要方法是证明每个可数虚自由群可嵌入到有限群的双群中。

Relating two genus 0 problems of John Thompson

The "relating" entwines three problems: 1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p. 2. Showing that the monodromy groups of rational function maps over the complexes are limited to a finite set of groups, outside of groups close to alternating groups (example, symmetric groups) with special representations, and dihedral and cyclic groups. 3. Relating the genus 0 modular curves to the character group of the Monster simple group, so-called Monstrous Moonshine. http://www.math.uci.edu/~mfried/pathlist-cov/thomp-genus0.html has …

通过球面的循环覆盖表示辫群：Zariski 闭包与算术性

Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity

本文研究了纯辫群通过平面曲线族纤维上同调的幺元在定义于有理数上的线性代数群中的本原表示，给出了其像的 Zariski 闭包为极大以及为算术格的判据，推广了前人结果并回答了 McMullen 的问题。