Quasimorphisms and laws

Aut-invariant quasimorphisms on graph products of abelian groups

The present paper constructs unbounded quasimorphisms that are invariant under all automorphisms on free products of more than two factors and on graph products of finitely generated abelian groups. This includes many classes of right angled Artin and right angled Coxeter groups. We discuss various geometrically arising families of graphs as examples and deduce the non-triviality of an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura for these groups.

Stable commutator length in word-hyperbolic groups

In this paper we obtain uniform positive lower bounds on stable commutator length in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on the Bass-Serre tree). If G is a word hyperbolic group which is delta hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on delta and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length is at least equal to …

关于准同态的扩展定理

An extension theorem for quasimorphisms

本文研究了双曲群的正规子群上准同态的扩展条件，并给出了稳定交换长度的双曲等价性结果。

Effective quasimorphisms on right-angled Artin groups

We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24. These results make use of some ne…

Bavard's duality theorem for mixed commutator length

Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-invariant if $f(gxg^{-1}) = f(x)$ for every $g \in G$ and every $x \in N$. The goal in this paper is to establish Bavard's duality theorem of $G$-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case $N = [G,N]$. Our duality theorem provides a connection between $G$-invariant quasimorphisms and $(G,N)$-commutator lengths. Here for $x \in [G,N]$, the $(G,N)$-commutator length $\mathrm{cl}_{G,N}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators which …

从字母拟态射到角结构与稳定换位子长度的谱间隙

From letter-quasimorphisms to angle structures and spectral gaps for scl

本文通过构造负曲率角结构，给出Heuer定理的新几何证明，证明在含字母拟态射的群（如RAAG）中稳定换位子长度下界为1/2，推广了自由群情形并联系Bavard对偶。

非可定向曲面微分同胚群上的拟态射

Quasimorphisms on nonorientable surface diffeomorphism groups

本文将可定向曲面微分同胚群上的拟态射构造推广至非可定向曲面，证明亏格至少为3的闭非可定向曲面微分同胚群单位分支上非平凡拟态射空间无穷维，并得出换位子长度与碎片长度无界。

不可扩张拟同态的空间

The space of non-extendable quasimorphisms

本研究针对群与正规子群对，构建相对有界子复形的上同调五项正合序列以刻画不可扩张拟同态空间，应用于多类群结构，并证明特定群对的稳定换位子长度与稳定混合换位子长度等价。

On stable commutator length in hyperelliptic mapping class groups

We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from ω-signatures, and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.

自由有理群上的稳定交换子长度

Stable commutator length on free $\mathbb{Q}$-groups

本文研究自由有理群上的稳定交换子长度，证明非单位元具有正长度，并展示自由有理群包含非定向曲面群的等距嵌入。

Biharmonic functions on groups and limit theorems for quasimorphisms along random walks

We show for very general classes of measures on locally compact second countable groups that every Borel measurable quasimorphism is at bounded distance from a quasi-biharmonic one. This allows us to deduce non-degenerate central limit theorems and laws of the iterated logarithm for such quasimorphisms along regular random walks on topological groups using classical martingale limit theorems of Billingsley and Stout. For quasi-biharmonic quasimorphism on countable groups we also obtain integral representations using martingale convergence.

Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group

These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a …

关于同胚群中的拟模与扭曲

On quasimorphisms and distortion in homeomorphism groups

本文研究了同胚群中的拟模与扭曲问题，识别了可扩展的拟模并探讨了其在群同调中的应用。

不变拟同态与广义混合Bavard对偶

Invariant quasimorphisms and generalized mixed Bavard duality

本文综述了Bavard对偶定理及其三种强化形式，提出新的广义混合Bavard对偶并给出统一自包含证明，还研究了与scl比较问题相关的不可扩张拟同态空间。

Gaps in scl for Amalgamated Free Products and RAAGs

We develop a new criterion to tell if a group $G$ has the maximal gap of $1/2$ in stable commutator length (scl). For amalgamated free products $G = A \star_C B$ we show that every element $g$ in the commutator subgroup of $G$ which does not conjugate into $A$ or $B$ satisfies $scl(g) \geq 1/2$, provided that $C$ embeds as a left relatively convex subgroup in both $A$ and $B$. We deduce from this that every non-trivial element $g$ in the commutator subgroup of a right-angled Artin group $G$ satisfies $scl(g) \geq 1/2$. This bound is sharp and is inherited by all fundamental groups of special c…

有限群同构、稳定交换子长度和固定点性质

Profinite isomorphisms, stable commutator length, and fixed point properties

本文通过构造Grothendieck对，证明稳定交换子长度、准同态等不是有限群不变量，并结合Rips构造与Dehn填充方法进行研究。

由Shelukhin拟同态构造的通量同态与双线性形式

Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

本文从Shelukhin拟同态出发，在闭连通辛流形的实一阶上同调上构造了辛映射类群不变的交替双线性形式，并由此给出可交换辛同胚通量的约束，推广了Rousseau关于开辛流形的工作。

Isometric endomorphisms of free groups

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolut…

稳定混合交换子长度的粗几何 I：对偶性与链上的泛函分析

Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains

本文研究了稳定混合交换子长度与普通稳定交换子长度之间的双利普希茨等价性问题，通过分析与不变拟同态相关的实线性空间W(G,N)的消失性，建立了泛函分析和链空间完成的理论框架。

$2$-orbifolds 中 scl 的一致谱间隙

Uniform spectral gap of scl in $2$-orbifolds

本文证明了所有紧致双曲 $2$-orbifolds 相对于外周子群的稳定换位子长度存在一致谱间隙，除三个锥点球面外，显式间隙为 $1/36$，方法包括显式拟态射和双曲几何（褶皱曲面）。

Constructing Double Magma with Commutation Operations

A double magma is a nonempty set with two binary operations satisfying the interchange law. We call a double magma proper if the two operations are distinct and commutative if the operations are commutative. A double semigroup is a double magma for which both operations are associative. Given a group G we define a double magma (G,*,#) with the commutator operations x * y = [x,y] (= x^-1y^-1xy) and x # y = [y,x]. We show that (G,*,#) is a double magma if and only if G satisfies the commutator laws [x,y;x,z]=1 and [w,x;y,z]^2 = 1. Note that the first law defines the variety of 3-metabelian group…

Positive laws on large sets of generators: counterexamples for infinitely generated groups

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.

稳定混合换位子长度的粗群理论研究

Coarse group theoretic study on stable mixed commutator length

本文利用粗群理论研究群及其正规子群上稳定混合换位子长度的大尺度行为，通过双不变度量将其与拟态商空间联系，证明了粗核同构于整数格，并给出了在有限生成群中的应用。

Uniformly perfect finitely generated simple left orderable groups

We show that the finitely generated simple left orderable groups $G_ρ$ constructed by the first two authors in arXiv:1807.06478 are uniformly perfect - each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any nontrivial action of the group on the circle, which lifts to an action on the real line, admits a fixed point. Most strikingly, it follows that the groups are examples of left orderable monsters, which means that any faithful action on the real line without…

群图与 $3$-流形中 scl 的谱间隙

Spectral gap of scl in graphs of groups and $3$-manifolds

本文研究树上群作用、群图与 $3$-流形基本群中的稳定交换子长度谱间隙，通过曲面映射刻画及拟态射与 Bavard 对偶性证明尖锐下界，并给出新例子。