scholarly journals Contractible, hyperbolic but non-CAT(0) complexes

2020 ◽  
Vol 30 (5) ◽  
pp. 1439-1463
Author(s):  
Richard C. H. Webb
Keyword(s):  
Isoperimetric Inequality ◽  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Index Subgroup ◽  
Analogous Statement ◽  
Almost All ◽  
Finite Index Subgroup

AbstractWe prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

scholarly journals The Casson–Walker invariant of 3-manifolds with genus one open book decompositions

2019 ◽  
Vol 28 (06) ◽  
pp. 1950018
Author(s):  
Atsushi Mochizuki
Keyword(s):  
Mapping Class Group ◽  
Central Extension ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Open Book ◽  
Open Book Decomposition ◽  
Open Book Decompositions ◽  
Framed Link ◽  
Genus One

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

scholarly journals The Johnson homomorphism and its kernel

2018 ◽  
Vol 2018 (735) ◽  
pp. 109-141 ◽  
Author(s):  
Andrew Putman
Keyword(s):  
Finite Index ◽  
General Result ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Finite Index Subgroups

AbstractWe give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

scholarly journals SIMPLY INTERSECTING PAIR MAPS IN THE MAPPING CLASS GROUP

2012 ◽  
Vol 21 (11) ◽  
pp. 1250107
Author(s):  
LEAH R. CHILDERS
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Infinite Index ◽  
Index Subgroup ◽  
Torelli Group ◽  
Topological Characterization

The Torelli group, [Formula: see text], is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying [Formula: see text]: bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on [Formula: see text], while SIP-maps have received relatively little attention until recently, due to an infinite presentation of [Formula: see text] introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP (Sg), and show it is an infinite index subgroup of [Formula: see text].

scholarly journals The extended mapping class group can be generated by two torsions

2017 ◽  
Vol 26 (11) ◽  
pp. 1750061
Author(s):  
Xiaoming Du
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Oriented Surface ◽  
Genus Formula ◽  
Extended Mapping ◽  
Closed Oriented Surface

Let [Formula: see text] be the closed-oriented surface of genus [Formula: see text] and let [Formula: see text] be the extended mapping class group of [Formula: see text]. When the genus is at least 5, we prove that [Formula: see text] can be generated by two torsion elements. One of these generators is of order [Formula: see text], and the other one is of order [Formula: see text].

scholarly journals Isomorphisms Between Big Mapping Class Groups

2018 ◽  
Vol 2020 (10) ◽  
pp. 3084-3099 ◽  
Author(s):  
Juliette Bavard ◽  
Spencer Dowdall ◽  
Kasra Rafi
Keyword(s):  
Finite Index ◽  
Outer Automorphism ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Index Subgroup ◽  
Class Groups ◽  
Infinite Type ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups ◽  
Orientable Surfaces

Abstract We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these “big” mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

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.

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