Surface Transformation Class Group

The group of classes of surface transformations is a group of homeomorphisms up to continuous deformation. It naturally arises in the study of three-dimensional manifolds and is associated with other groups, in particular with braid groups and the group of external automorphisms of the group.

A group of classes of mappings can be defined for arbitrary varieties and for arbitrary topological spaces, but the case of surfaces is the most studied in group theory .

Content

History

The study of groups of classes of mappings was initiated by Max Den and . Den constructed a finite system of generators of this group, ^[1] and Nielsen proved that all automorphisms of the fundamental surface groups are initiated by homeomorphisms.

In the mid-seventies, William Thurston used this group to study three-dimensional manifolds. ^[2]

Later, a group of classes began to be studied in the geometric theory of groups , where it serves as a testing ground for various hypotheses and the development of technical tools.

Definition

Let be $S$ ${\ displaystyle S}$ there is a connected , closed , orientable surface, and $\mathrm {Homeo} ^{+}(S)$ ${\ displaystyle \ mathrm {Homeo} ^ {+} (S)}$ there is a group of orientation-preserving homeomorphisms equipped with a compact-open topology .

Connected component unit in $\mathrm {Homeo} ^{+}(S)$ ${\ displaystyle \ mathrm {Homeo} ^ {+} (S)}$ is indicated $\mathrm {Homeo} _{0}(S)$ ${\ displaystyle \ mathrm {Homeo} _ {0} (S)}$ . It consists of homeomorphisms $\mathrm {Homeo} ^{+}(S)$ ${\ displaystyle \ mathrm {Homeo} ^ {+} (S)}$ isotopic to the identity homeomorphism. Subgroup $\mathrm {Homeo} _{0}(S)$ ${\ displaystyle \ mathrm {Homeo} _ {0} (S)}$ is a normal subgroup $\mathrm {Homeo} ^{+}(S)$ ${\ displaystyle \ mathrm {Homeo} ^ {+} (S)}$ .

A group of transformation surface classes of mappings $S$ ${\ displaystyle S}$ defined as a factor group

\mathrm {Mod} (S)=\mathrm {Homeo} ^{+}(S)/\mathrm {Homeo} _{0}(S).

{\ displaystyle \ mathrm {Mod} (S) = \ mathrm {Homeo} ^ {+} (S) / \ mathrm {Homeo} _ {0} (S).}

Remarks

If in this definition we use all homeomorphisms (not only preserving orientation), we obtain an extended group of transformation classes $\mathrm {Mod} ^{\pm }(S)$ ${\ displaystyle \ mathrm {Mod} ^ {\ pm} (S)}$ in which the group $\mathrm {Mod} (S)$ ${\ displaystyle \ mathrm {Mod} (S)}$ contained as a subgroup of index 2.

This definition can also be given for the category of diffeomorphisms . More precisely, if the word "homeomorphism" is replaced everywhere with " diffeomorphism ", we get the same group, since the inclusion $\mathrm {Diff} ^{+}(S)\subset \mathrm {Homeo} ^{+}(S)$ ${\ displaystyle \ mathrm {Diff} ^ {+} (S) \ subset \ mathrm {Homeo} ^ {+} (S)}$ induces isomorphism by the corresponding classes.

In the case when $S$ ${\ displaystyle S}$ - compact surface with an edge $\partial S$ ${\ displaystyle \ partial S}$ , in the definition we take only homeomorphisms fixing all points on the edge.

For punctured surfaces, the group is defined exactly as described above.
- Note that class mapping is allowed to rearrange punctured points, but not edge components.

Examples

The group of transformation classes of the sphere is trivial.
Torus mapping class group $\mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}$ ${\ displaystyle \ mathbb {T} ^ {2} = \ mathbb {R} ^ {2} / \ mathbb {Z} ^ {2}}$ naturally isomorphic to the modular group $\mathrm {SL} _{2}(\mathbb {Z} )$ ${\ displaystyle \ mathrm {SL} _ {2} (\ mathbb {Z})}$ .
The class group of ring mappings is a cyclic group formed by one Dehn twist .
The braid group with n threads is naturally isomorphic to the group of transformation classes of the disk with n punctured points.

Properties

The class group of surface transformations is countable .

An extended group of transformation classes of a surface without boundary is isomorphic to the automorphism group of its fundamental group.
- Moreover, any automorphism of the fundamental group is induced by some surface homeomorphism.
- Generally speaking, the statement ceases to be true for surfaces with an edge. In this case, the fundamental group is a free group, and the group of external automorphisms of the group includes the group of classes of surface transformations as a proper subgroup.

For compact surface $S$ ${\ displaystyle S}$ and $x\in S$ ${\ displaystyle x \ in S}$ exact sequence exists
$1\to \pi _{1}(S,x)\to \mathrm {Mod} (S\setminus \{x\})\to \mathrm {Mod} (S)\to 1.$ ${\ displaystyle 1 \ to \ pi _ {1} (S, x) \ to \ mathrm {Mod} (S \ setminus \ {x \}) \ to \ mathrm {Mod} (S) \ to 1.}$

Any item $α$ {\ displaystyle \ alpha} Groups of classes of surface transformations fall into one of three categories:
- $\alpha$ ${\ displaystyle \ alpha}$ has a finite order (i.e. $\alpha ^{n}=1$ ${\ displaystyle \ alpha ^ {n} = 1}$ for some $n$ ${\ displaystyle n}$ );
- $\alpha$ ${\ displaystyle \ alpha}$ reducible, that is, there exists a set of disjoint closed curves on $S$ ${\ displaystyle S}$ persisting under the action of $\alpha$ ${\ displaystyle \ alpha}$ ;
- $\alpha$ ${\ displaystyle \ alpha}$ .

A group of classes of surface transformations can be generated
- Two elements ^[3]
- Involutions ^[4]
- There is a finite assignment with Dan twists as generators.
  - The smallest number of Dehn twists forming a group of classes of transformations of the genus surface $g\geqslant 2$ ${\ displaystyle g \ geqslant 2}$ equals $2g+1$ ${\ displaystyle 2g + 1}$ .

A group of classes of surface transformations naturally acts on its Teichmüller space .
- This action is actually discontinuous , not free.
- Metrics on the Teichmüller space can be used to establish some global properties of a group of transformation classes. For example, from this it follows that a maximal quasi-isometrically embedded plane in a group of transformation classes of a surface of genus $g$ ${\ displaystyle g}$ have dimension $3g-3+k$ ${\ displaystyle 3g-3 + k}$ . ^[5]

The group of classes of surface transformations naturally acts on the surface. This action, together with the combinatorial-geometric properties of a complex of curves, can be used to prove various properties of a group of transformation classes.
- In particular, this explains the group’s presence of some properties close to Gromov’s hyperbolicity .

The first homologies of the group of classes of surface transformations are finite.
- It follows from this that the first cohomology groups are also finite.

The group of classes of surface transformations is residually finite .

The class group of surface transformations has only a finite number of conjugacy classes.

It is not known whether the group of classes of surface transformations is a linear group. In addition to symplectic representations on homology, other linear representations arising from topological quantum field theory are also known. The images of these representations are contained in arithmetic groups that are not symplectic ^[6] .
Dimension of nontrivial action of a group of classes of transformations of a surface of genus $g$ ${\ displaystyle g}$ cannot be less $2{\sqrt {g-1}}$ ${\ displaystyle 2 {\ sqrt {g-1}}}$ ^[7] .

Notes

↑ Dehn, Max. Die Gruppe de Abbildungsklassen (neopr.) // Acta Mathematica . - 1938.- T. 69 . - S. 135-206 . - DOI : 10.1007 / bf02547712 .
↑ Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces (English) // Bull. Amer. Math. Soc. : journal. - 1988. - Vol. 19 . - P. 417-431 . - DOI : 10.1090 / s0273-0979-1988-15685-6 .
↑ Wajnryb, B. Mapping class group of a surface is generated by two elements (Eng.) // Topology: journal. - 1996. - Vol. 35 . - P. 377-383 . - DOI : 10.1016 / 0040-9383 (95) 00037-2 .
↑ Tara E. Brendle, Benson Farb. Every mapping class group is generated by 3 torsion elements and by 6 involutions (English) // J. Algebra: journal. - 2004. - Vol. 278 . MR : 187C198
↑ Alex Eskin, Howard Masur, Kasra Rafi (2014), "Large scale rank of Teichmüller space", arΧiv : 1307.3733 [math.GT] .
↑ Masbaum, Gregor and Reid, Alan W. All finite groups are involved in the mapping class group (Eng.) // Geom. Topol. : journal. - 2012. - Vol. 16 . - P. 1393-1411 . - DOI : 10.2140 / gt.2012.16.1393 . MR : 2967055
↑ Benson Farb, Alexander Lubotzky, Yair Minsky. Rank-1 phenomena for mapping class groups (neopr.) // Duke Math. J. . - 2001 .-- T. 106 . - S. 581-597 . MR : 1813237

Literature

Alexey Zhirov. Topological conjugation of pseudo-Anosov homeomorphisms. - MCCMO, 2013 .-- 1000 copies. - ISBN 978-5-4439-0213-5 .
AA Gayfulin , Groups of classes of mappings and their subgroups .

[1] Dehn, Max. Die Gruppe de Abbildungsklassen (neopr.) // Acta Mathematica . - 1938.- T. 69 . - S. 135-206 . - DOI : 10.1007 / bf02547712 .

[2] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces (English) // Bull. Amer. Math. Soc. : journal. - 1988. - Vol. 19 . - P. 417-431 . - DOI : 10.1090 / s0273-0979-1988-15685-6 .

[3] Wajnryb, B. Mapping class group of a surface is generated by two elements (Eng.) // Topology: journal. - 1996. - Vol. 35 . - P. 377-383 . - DOI : 10.1016 / 0040-9383 (95) 00037-2 .

[4] Tara E. Brendle, Benson Farb. Every mapping class group is generated by 3 torsion elements and by 6 involutions (English) // J. Algebra: journal. - 2004. - Vol. 278 . MR : 187C198

[5] Alex Eskin, Howard Masur, Kasra Rafi (2014), "Large scale rank of Teichmüller space", arΧiv : 1307.3733 [math.GT] .

[6] Masbaum, Gregor and Reid, Alan W. All finite groups are involved in the mapping class group (Eng.) // Geom. Topol. : journal. - 2012. - Vol. 16 . - P. 1393-1411 . - DOI : 10.2140 / gt.2012.16.1393 . MR : 2967055

[7] Benson Farb, Alexander Lubotzky, Yair Minsky. Rank-1 phenomena for mapping class groups (neopr.) // Duke Math. J. . - 2001 .-- T. 106 . - S. 581-597 . MR : 1813237