scholarly journals Cubical geometry in the polygonalisation complex

2018 ◽  
Vol 167 (01) ◽  
pp. 1-22
Author(s):  
MARK C. BELL ◽  
VALENTINA DISARLO ◽  
ROBERT TANG
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Geometric Model ◽  
Mapping Class ◽  
Cube Complex ◽  
Flip Graph ◽  
Extended Mapping

AbstractWe introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface.We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov's link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.

scholarly journals Mapping class groups of highly connected $$(4k+2)$$-manifolds

2020 ◽  
Vol 26 (5) ◽  
Author(s):  
Manuel Krannich
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups

AbstractWe compute the mapping class group of the manifolds $$\sharp ^g(S^{2k+1}\times S^{2k+1})$$ ♯ g ( S 2 k + 1 × S 2 k + 1 ) for $$k>0$$ k > 0 in terms of the automorphism group of the middle homology and the group of homotopy $$(4k+3)$$ ( 4 k + 3 ) -spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.

scholarly journals Automorphisms of braid groups on orientable surfaces

2016 ◽  
Vol 25 (05) ◽  
pp. 1650022
Author(s):  
Byung Hee An
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Orientable Surface ◽  
Braid Groups ◽  
Mapping Class ◽  
Characteristic Subgroup ◽  
Group Extensions ◽  
Extended Mapping ◽  
Orientable Surfaces

In this paper, we compute the automorphism groups [Formula: see text] and [Formula: see text] of braid groups [Formula: see text] and [Formula: see text] on every orientable surface [Formula: see text], which are isomorphic to group extensions of the extended mapping class group [Formula: see text] by the transvection subgroup except for a few cases. We also prove that [Formula: see text] is always a characteristic subgroup of [Formula: see text], unless [Formula: see text] is a twice-punctured sphere and [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 On the arc and curve complex of a surface

2009 ◽  
Vol 148 (3) ◽  
pp. 473-483 ◽  
Author(s):  
MUSTAFA KORKMAZ ◽  
ATHANASE PAPADOPOULOS
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Combinatorial Structure ◽  
Mapping Class ◽  
Natural Image ◽  
Curve Complex ◽  
Natural Embedding ◽  
Connected Surface ◽  
Gromov Hyperbolic

AbstractWe study the arc and curve complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the extended mapping class group of S in that group. We also show that for any vertex of AC(S), the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC(S) is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on S, was already known. As a corollary, AC(S) is Gromov-hyperbolic.

scholarly journals EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS

2013 ◽  
Vol 05 (01) ◽  
pp. 57-85 ◽  
Author(s):  
MATTHEW B. DAY
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Lie Algebras ◽  
Mapping Class Group ◽  
Finite Rank ◽  
Class Group ◽  
Homogeneous Spaces ◽  
Group Cohomology ◽  
Mapping Class ◽  
Nilpotent Lie Algebras

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [5]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.

scholarly journals Separating subgroups of mapping class groups in homological representations

2020 ◽  
pp. 1-15
Author(s):  
Asaf Hadari
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Representation Formula ◽  
Closed Surface ◽  
Mapping Class ◽  
Finite Cover ◽  
Class Groups ◽  
Genus Formula

Let [Formula: see text] be either the mapping class group of a closed surface of genus [Formula: see text], or the automorphism group of a free group of rank [Formula: see text]. Given any homological representation [Formula: see text] of [Formula: see text] corresponding to a finite cover, and any term [Formula: see text] of the Johnson filtration, we show that [Formula: see text] has finite index in [Formula: see text], the Torelli subgroup of [Formula: see text]. Since [Formula: see text] for [Formula: see text], this implies for instance that no such representation is faithful.

AUTOMORPHISMS OF BRAID GROUPS ON CLOSED SURFACES WHICH ARE NOT S2, T2, P2 OR THE KLEIN BOTTLE

2006 ◽  
Vol 15 (09) ◽  
pp. 1231-1244 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Outer Automorphism ◽  
Closed Surface ◽  
Braid Groups ◽  
Mapping Class ◽  
Closed Surfaces ◽  
Extended Mapping

Consider a surface braid group of n strings as a subgroup of the isotopy group of homeomorphisms of the surface permuting n fixed distinguished points. Each automorphism of the surface braid group (respectively, of the special surface braid group) is shown to be a conjugate action on the braid group (respectively, on the special braid group) induced by a homeomorphism of the underlying surface if the closed surface, either orientable or non-orientable, is of negative Euler characteristic. In other words, the group of automorphisms of such a surface braid group is isomorphic to the extended mapping class group of the surface with n punctures, while the outer automorphism group of the surface braid group is isomorphic to the extended mapping class group of the closed surface itself.

scholarly journals OUTER AUTOMORPHISMS OF MAPPING CLASS GROUPS OF NONORIENTABLE SURFACES

2010 ◽  
Vol 20 (03) ◽  
pp. 437-456 ◽  
Author(s):  
FERIHE ATALAN
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Outer Automorphism ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Nonorientable Surface ◽  
Nonorientable Surfaces ◽  
Outer Automorphisms

Let Ng be the connected closed nonorientable surface of genus g ≥ 5 and Mod (Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod (Ng) is cyclic.

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