Abelian subgroups of the mapping class groups for non-orientable surfaces
Erika Kuno
Tokyo Institute of Technology (Research Fellow of JSPS DC2)
December 23, 2016＠Nihon University
1... Introduction
2... Preliminaries
3... Main results
4... Differences from orientable surfaces
Introduction
S =Sg,n: a compact connected orientable surface of genus gwith n boundary components s.t. χ(S)<0.
N =Ng,n: a compact connected non-orientable surface of genus g with nboundary components s.t. χ(N)<0.
F =S or N.
M(F): the mapping class group of F,
i.e. the group of isotopy classes of (orientation preserving ifF =S) self-homeomorphisms of F with isotopies fixing each boundary component of F setwise.
.Theorem (Birman-Lubotzky-McCarthy 1983) ..
...
G: any abelian subgroup of M(S).
Then Gis finitely generated and the torsion-free rank ofG is bounded by 3g+n−3.
Idea of proof To show
rank(G)≤cardinality{Dehn twists along pairwise disjoint curves on S}.
Introduction
Double covering map of a non-orientable surface N.
This double covering map induces an injective homomorphism ι:M(Ng,n)→M(Sg−1,2n).
.Corollary ..
...
G: any abelian subgroup of M(Ng,n).
Then Gis finitely generated and the torsion-free rank ofG is bounded by 3(g−1) + 2n−3.
−→ This bound might not be best possible.
Introduction
Moreover
.Theorem (Szepietowski 2010) ..
...
Any Dehn twists are not contained inι(M(Ng,n)).
−→ No Dehn twists in M(Sg−1,2n) have lifts in M(Ng,n) byι.
−→ We don’t know the maximal torsion-free rank of the abelian
subgroups of M(N) by the result of Birman-Lubotzky-McCarthy directly.
Q.
What is the maximal rank of the abelian subgroups of M(N)?
Moreover
.Theorem (Szepietowski 2010) ..
...
Any Dehn twists are not contained inι(M(Ng,n)).
−→ No Dehn twists in M(Sg−1,2n) have lifts in M(Ng,n) byι.
−→ We don’t know the maximal torsion-free rank of the abelian
subgroups of M(N) by the result of Birman-Lubotzky-McCarthy directly.
Q.
What is the maximal rank of the abelian subgroups of M(N)?
Introduction
Moreover
.Theorem (Szepietowski 2010) ..
...
Any Dehn twists are not contained inι(M(Ng,n)).
−→ No Dehn twists in M(Sg−1,2n) have lifts in M(Ng,n) byι.
−→ We don’t know the maximal torsion-free rank of the abelian
subgroups of M(N) by the result of Birman-Lubotzky-McCarthy directly.
Q.
What is the maximal rank of the abelian subgroups of M(N)?
The following two theorems gave partial answers of the question.
.Theorem (Atalan-Szepietowski 2014) ..
...
N: a non-orientable surface of odd genus g≥5.
Then the maximal rank of the abelian subgroups of M(N)is
3
2(g−1) +n−2.
.Theorem (Atalan 2015) ..
...
N: a non-orientable surface of even genus.
The maximal rank of the abelian subgroups ofM(N), which contain a group generated by Dehn twists is 32g+n−3.
−→ We give the answer of this question!
Introduction
The following two theorems gave partial answers of the question.
.Theorem (Atalan-Szepietowski 2014) ..
...
N: a non-orientable surface of odd genus g≥5.
Then the maximal rank of the abelian subgroups of M(N)is
3
2(g−1) +n−2.
.Theorem (Atalan 2015) ..
...
N: a non-orientable surface of even genus.
The maximal rank of the abelian subgroups ofM(N), which contain a group generated by Dehn twists is 32g+n−3.
−→ We give the answer of this question!
Appendix
.Theorem (Ivanov 2016) ..
...
S: a closed orientable surface.
I(S): the Torelli group of S.
Every abelian subgroup ofI(S) is a free abelian group with rank bounded by2g−3.
Preliminaries
⋄ N =Ng,n: a non-orientable surface of genus g withn boundary components:
⋄ arcs on N:
properly embedded and essential , i.e. are not isotopic into ∂N
⋄ curves on N:
properly embedded and essential ,
i.e. do not bound a disk or a M¨obius band, and are not isotopic to ∂N
Preliminaries
⋄ a one-sided curve a: the regular neighborhood ofain N is a M¨obius band.
⋄ a two-sided curve a: the regular neighborhood of ain N is an annulus.
⋄ A Dehn twist ta along a two-sided curve ain N:
Preliminaries
⋄ φ∈M(N) is reducible if∃f ∈φand ∃A: a family of curves s.t.
f(A) =A. (We call such systemsreduction systems for f.)
⋄ φ∈M(N) is of finite order if∃f ∈φand ∃n̸= 0 s.t. fn= idN.
⋄ φ∈M(N) is pseudo-Anosov if ∃f ∈φ s.t. ∀a: a curve and ∀n̸= 0, fn(a)̸=a.
.Theorem 1 (K.) ..
...
N: a non-orientable surface with χ(N)<0.
G: a torsion-free abelian subgroup ofM(N).
Then, G∼=⟨τ1,· · · , τk⟩<M(N), where each τi is an isotopy class of a Dehn twist and the supports of τi andτj are disjoint for i̸=j.
Further, k≤ 32(g−1) +n−2 ifg is odd and k≤ 32g+n−3 ifg is even.
odd genus even genus
Differences from orientable surfaces
A: a set of isotopy classes of curves whose representatives can be chosen to consist of pairwise disjoint.
A: a set of the representatives ofA which are mutually disjoint.
NA: the natural compactification ofN −A.
MA(N): the stabilizer ofA in M(N).
−→ We can define a well-defined homomorphism Λ :MA(N)→M(NA) as follows:
A: a set of isotopy classes of curves whose representatives can be chosen to consist of pairwise disjoint.
A: a set of the representatives ofA which are mutually disjoint.
NA: the natural compactification ofN −A.
MA(N): the stabilizer ofA in M(N).
−→ We can define a well-defined homomorphism Λ :MA(N)→M(NA) as follows:
Differences from orientable surfaces
For φ∈MA(N),
we can choose a set Aof the representatives of A andf ∈φs.t.
f(A) =A.
Further, f|N−A extends uniquely toNA (we put itfˆ).
−→ This process determines a well-defined class φˆ= [ ˆf]∈M(NA).
Atwo: the set of all isotopy classes of two-sided curves inA. α∈Atwo.
a: a representative ofα.
ta: the Dehn twist alonga.
τα: the isotopy class ofta. Def.
A reduction system A for φ∈M(N) is adequateif each of restrictions of φ to each component ofNA is either of finite order or pseudo-Anosov.
Differences from orientable surfaces
One of the differences is the following: Λ : MA(N)→M(NA)
Lemma
Ker(Λ) =⟨τα |α∈Atwo⟩.
The second difference is the following: Lemma
F: a cpt. conn. surf. with χ(F)<0.
δ: any isotopy class of properly embedded arc onF. φ∈M(F)with φ(δ) =δ.
Then one of the following occurs.
(1) F =S0,3 or N1,2 or N2,1.
Differences from orientable surfaces
The second difference is the following: Lemma
F: a cpt. conn. surf. with χ(F)<0.
δ: any isotopy class of properly embedded arc onF. φ∈M(F)with φ(δ) =δ.
Then one of the following occurs.
(1) F =S0,3 or N1,2 or N2,1.
(2) If δ connects two distinct boundary components
−→ ∃γ: an isotopy class of a curve s.t. φ(γ) =γ and i(α, γ)̸= 0 for∀α:
an isotopy class of a curve with i(α, δ)̸= 0.
Differences from orientable surfaces
(3) If δ is an isotopy class of an arc which connects one boundary
component, goes through crosscaps even number of times, and surrounds one crosscap
−→ for anyα excepting β0 with i(α, δ)̸= 0 there exists γ such that φ(γ) =γ andi(α, γ)̸= 0.
(4) If δ is an isotopy class of an arc which connects one boundary component, goes through crosscaps even number of times, and does not surround one crosscap
−→ for anyα with i(α, δ)̸= 0 there existsγ such that φ(γ) =γ and i(α, γ)̸= 0.
Differences from orientable surfaces
(5) If δ is an isotopy class of an arc which connects one boundary component, goes through crosscaps odd number of times
−→ for anyα excepting β1 andβ2 with i(α, δ)̸= 0 there exists γ such that φ(γ) =γ and i(α, γ)̸= 0.
The result of the second lemma is different from that of the orientable surfaces.
However
the result of the following lemma is the same as that of the orientable surfaces:
Differences from orientable surfaces
The result of the second lemma is different from that of the orientable surfaces.
However
the result of the following lemma is the same as that of the orientable surfaces:
.Lemma ..
...
A: an adequate reduction system forφ∈M(N).
α∈A.
A′ =A − {α}.
Then, the following are equivalent.
(1) α is an essential reduction class, where a reduction classα for φis essential if for any isotopy class of a curveβ such that i(α, β)̸= 0 and m̸= 0, the classφm(β)̸=β.
(2) A′ is not an adequate reduction system for φm for anym̸= 0.
−→ We can use Birman-Lubotzky-McCarthy’s techniques to the non-orientable surfaces.
Differences from orientable surfaces
.Lemma ..
...
A: an adequate reduction system forφ∈M(N).
α∈A.
A′ =A − {α}.
Then, the following are equivalent.
(1) α is an essential reduction class, where a reduction classα for φis essential if for any isotopy class of a curveβ such that i(α, β)̸= 0 and m̸= 0, the classφm(β)̸=β.
(2) A′ is not an adequate reduction system for φm for anym̸= 0.
−→ We can use Birman-Lubotzky-McCarthy’s techniques to the non-orientable surfaces.
