Related papers: Stable W-length

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…

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov…

Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined.…

Let $w$ be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients…

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high…

We prove that for any euclidean ring R and n at least 6, Gamma=SL_n(R) has no unbounded quasi-homomorphisms. From Bavard's duality theorem, this means that the stable commutator length vanishes on Gamma. The result is particularly…

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities…

We study the bounded cohomology and the stable commutator length of verbal wreath products $\Gamma \wr^{_W}A$, where $A$ has trivial bounded cohomology for a sufficiently large class of coefficients.\\ We prove that the stable commutator…

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…

Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu_{w,G}$ on $G$ by (Haar-)uniformly sampling $g_1,...,g_r\in G$ and evaluating $w(g_1,...,g_r)$. In [arXiv:1802.04862], Magee…

This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is…

Stable commutator length scl_G(g) of an element g in a group G is an invariant for group elements sensitive to the geometry and dynamics of G. For any group G acting on a tree, we prove a sharp bound scl_G(g)>=1/2 for any g acting without…

The width $\wid(G,W)$ of the verbal subgroup $v(G,W)$ of a group $G$ defined by a collection of group words $W$ is the smallest number $m$ in $\mathbb N \cup {+\infty}$ such that every element of $v(G,W)$ is can be represented as the…

For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G…

A homogeneous quasimorphism $\phi$ on a normal subgroup $N$ of $G$ is said to be $G$-invariant if $\phi(gxg^{-1}) = \phi(x)$ for every $g \in G$ and for every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic…

We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and…

This paper presents a simplification of the main argument in "Effective quasimorphisms on right-angled Artin groups" by Fern\'os, Forester and Tao. Their article introduces a family of quasimorphisms on a certain class of groups (called…

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal…

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for…

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all…