On the behaviour of the Atiyah Conjecture under taking subgroups and under taking quotients with finite kernel

Quotients and subgroups of Baumslag-Solitar groups

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).

The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n/[P_n,P_n]$

Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.

On power subgroups of mapping class groups

In the first part of this paper we prove that the mapping class subgroups generated by the $D$-th powers of Dehn twists (with $D\geq 2$) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal subgroup generated by the $D$-th powers of all elements of the mapping class groups. We show first that for infinitely many $D$ the power quotient groups are non-trivial. On the other hand, if $4g+2$ does not divide $D$ then the associated power quotient of the mapping class group…

辫群模其同余子群的商群

Quotients of braid groups by their congruence subgroups

该研究解决了Dan Margalit提出的公开问题，确定了辫群同余子群在整一般线性群中的像，并利用辛同余子群刻画了辫群模同余子群的商群结构。

Hyperbolic quotients of projection complexes

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $δ$-hyperbolic. Additionally, under certain circumstan…

辫群的商群：对称群的扩张

Quotients of the braid group that are extensions of the symmetric group

本文研究辫群中使商为对称群阿贝尔扩张的正规子群，证明其恰有8个可公度类，定义Specht子群并刻画其模与上同调性质，确定扩张是否分裂。

薄正规子群的共轭化子与阿贝尔商

Commensurators of thin normal subgroups and abelian quotients

本文证明了秩一李群中算术格的无限正规子群，若其商群有到整数群的满同态，则该子群的共轭化子是离散的，并推广了先前关于PSL2(Z)的结果。

Quotients of the Artin braid groups and crystallographic groups

Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class …

Abelian quotients of subgroups of the mapping class group and higher Prym representations

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the cla…

对称群虚拟扩张的换位子群与晶体学商群

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

本文研究对称群虚拟扩张的换位子群与晶体学商群，给出纯虚拟三重群及两个换位子群的显式有限表示，并证明 $VL_n/PVL_n^{'}$ 为晶体学群且构造了无穷多个挠元素。

Distinguishing geometries using finite quotients

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for wh…

小 Coxeter 群的全等子群与晶体学商群

Congruence subgroups and crystallographic quotients of small Coxeter groups

本文研究小 Coxeter 群的全等子群性质，证明无限且非 virtually abelian 的小 Coxeter 群不满足该性质，并确定孪生群与三重群的晶体学商群及其维数和 holonomy 表示。

Integrality of L2-Betti numbers

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new class of groups for which the Atiyah conjecture holds, which contains all free groups and in par…

Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of the two families have all $2$-cusps, so that we can saturate the entire volume spectrum with $2$-cusped manifolds. Finally, we show that the associated neat lattices have infinite abelianization and finitely generated commut…

Finite quotients of braid groups

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.

Controlled embeddings into groups that have no non-trivial finite quotients

If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isometric embedding into a compact, connected, non-positively curved space Overline(X) such that Overline(X) has no non-trivial finite-sheeted coverings.

Alternating quotients of right-angled Coxeter groups

Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ γ_1, \ldots , γ_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (γ_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely gene…

映射类群关于幂子群的商群

Quotients of the mapping class group by power subgroups

本文研究映射类群关于 Dehn 扭转幂子群的商群，发现其包含无限指标正规子群或 Kähler 子群，并与有限指标子群的阿贝尔化问题等价。

Arithmetic quotients of the mapping class group

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $τ$. To this pair $(A,τ)$, we associate an arithmetic group $Ω$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian sesquilinear form on $A^{2g-2}$. We show that if $H$ is generated by less than $g$ elements, then $Ω$ is a virtual quotient of the mapping class group $Mod(Σ_g)$, i.e. a finite index subgroup of $Ω$ is a quotient of a finite index subgroup of $\Mod(Σ_g)$. This shows that the mapp…

Discrete groups without finite quotients

We construct an infinite discrete subgroup of the isometry group of $\mathbb H^3$ with no finite quotients other than the trivial group.

On the quotient of the homology cobordism group by Seifert spaces

We prove that the quotient of the integer homology cobordism group by the subgroup generated by the Seifert fibered spaces is infinitely generated.

Small Quotients of Braid Groups

We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is the smallest non-trivial quotient of the commutator subgroup $B_n'$ for $n = 5,6,7,8$. We also give an improved lower bound on the order of any non-cyclic quotient of $B_n$.

McKay's correspondence for cocompact discrete subgroups of SU(1,1)

The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replaced with a cocompact discrete subgroup of the universal cover of SU(1,1) such that its image in PSU(1,1) is a cocompact fuchsian group with quotient of genus 0. We establish a correspondence between a certain class of finite-dimensional unitary representations of G and vector bundl…