scholarly journals Concordance group and stable commutator length in braid groups

2015 ◽  
Vol 15 (5) ◽  
pp. 2861-2886 ◽  
Author(s):  
Michael Brandenbursky ◽  
Jarek Kędra
Keyword(s):  
Braid Groups ◽  
Stable Commutator Length ◽  
Commutator Length

scholarly journals Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

scholarly journals Stable commutator length in amalgamated free products

2015 ◽  
Vol 07 (04) ◽  
pp. 693-717 ◽  
Author(s):  
Tim Susse
Keyword(s):  
Geometric Model ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Torus Knot ◽  
Knot Complements ◽  
Stable Commutator Length ◽  
Free Abelian Groups ◽  
Commutator Length ◽  
Boundary Mapping

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

scholarly journals On stable commutator length in hyperelliptic mapping class groups

2014 ◽  
Vol 272 (2) ◽  
pp. 323-351 ◽  
Author(s):  
Danny Calegari ◽  
Naoyuki Monden ◽  
Masatoshi Sato
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Stable Commutator Length ◽  
Commutator Length

scholarly journals FINITELY GENERATED INFINITE SIMPLE GROUPS OF INFINITE SQUARE WIDTH AND VANISHING STABLE COMMUTATOR LENGTH

2010 ◽  
Vol 02 (03) ◽  
pp. 341-384 ◽  
Author(s):  
ALEXEY MURANOV
Keyword(s):  
Simple Groups ◽  
Finitely Generated ◽  
Invariant Norm ◽  
Stable Commutator Length ◽  
Commutator Length ◽  
Commutator Width

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

scholarly journals Statistics and compression of scl

2014 ◽  
Vol 35 (1) ◽  
pp. 64-110 ◽  
Author(s):  
DANNY CALEGARI ◽  
JOSEPH MAHER
Keyword(s):  
Random Walk ◽  
Outer Automorphism ◽  
Hyperbolic Spaces ◽  
Mapping Class ◽  
Sharp Estimates ◽  
Stable Commutator Length ◽  
Commutator Length ◽  
Random Subspaces ◽  
Normed Vector ◽  
Translation Length

AbstractWe 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 non-degenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length$n$is of order$n/ \log n$. This establishes quantitative refinements of qualitative results of Bestvina and Fujiwara and others on the infinite dimensionality of two-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length$n$in a mapping class group cannot be written as a product of fewer than$O(n/ \log n)$reducible elements, with probability going to$1$as$n$goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length$n$on the outer automorphism group of a free group grows linearly in$n$.

scholarly journals Aut-invariant quasimorphisms on free products

2021 ◽  
Author(s):  
Bastien Karlhofer
Keyword(s):  
Free Product ◽  
Dihedral Group ◽  
Free Products ◽  
Infinite Dimensional ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractLet $$G=A *B$$ G = A ∗ B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

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