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 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 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 Stable commutator length on mapping class groups

2016 ◽  
Vol 66 (3) ◽  
pp. 871-898 ◽  
Author(s):  
Mladen Bestvina ◽  
Ken Bromberg ◽  
Koji Fujiwara
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Stable Commutator Length ◽  
Commutator Length

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

scholarly journals Cogrowth for group actions with strongly contracting elements

2018 ◽  
Vol 40 (7) ◽  
pp. 1738-1754 ◽  
Author(s):  
GOULNARA N. ARZHANTSEVA ◽  
CHRISTOPHER H. CASHEN
Keyword(s):  
Mapping Class Groups ◽  
Hyperbolic Spaces ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Small Cancellation ◽  
Class Groups ◽  
Original Result ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Right Angled Artin Groups

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

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.

The Dehn-Nielsen-Baer Theorem

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Harmonic Maps ◽  
Outer Automorphism ◽  
Mapping Class ◽  
Original Proof ◽  
Outer Automorphism Groups ◽  
And Algebra ◽  
Manifold Theory

This chapter deals with the Dehn–Nielsen–Baer theorem, one of the most beautiful connections between topology and algebra in the mapping class group. It begins by defining the objects in the statement of the Dehn–Nielsen–Baer theorem, including the extended mapping class group and outer automorphism groups. It then considers the use of the notion of quasi-isometry in Dehn's original proof of the Dehn–Nielsen–Baer theorem. In particular, it discusses a theorem on the fundamental observation of geometric group theory, along with the property of being linked at infinity. It also presents the proof of the Dehn–Nielsen–Baer theorem and an analysis of the induced homeomorphism at infinity before concluding with two other proofs of the Dehn–Nielsen–Baer theorem, one inspired by 3-manifold theory and one using harmonic maps.

On the finite generation of high dimensional cohomology ring of virtually torsion-free groups

1998 ◽  
Vol 124 (1) ◽  
pp. 57-72 ◽  
Author(s):  
CHUN-NIP LEE
Keyword(s):  
Automorphism Groups ◽  
Cohomology Ring ◽  
Free Groups ◽  
Outer Automorphism ◽  
Cohomological Dimension ◽  
Group Cohomology ◽  
Mapping Class ◽  
Finitely Generated ◽  
Finite Generation ◽  
A Finite Group

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

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