Mathematics Problem Archive

(orderability). Does Modg,g ≥ 2 have some finite index subgroup which acts faithfully by homeomorphisms on S1? Does either Modg or Modg,1 have a finit...

((Non)residual finiteness). Is the (universal) central extension ˜Modg,1 of Modg,1 residually finite, or not? Note that an old result of Grossman stat...

(The sections problem). Determine those subgroups H ≤ Modg for which π has a section over H. Do this as well with Homeo+(Σg) replaced by various subgr...

(Sections over finite index subgroups). Does the natural map Homeo+(Σg)→ Modg have a section over a finite index subgroup of Modg, or not? Of course t...

Does Modg or any of its finite index subgroups have any faithful action by homeomorphisms on Σg?...

Does limg→∞gL(Modg) exist? Another basic open question is the following....

Is the sequence {L(Modg)} monotone decreasing? strictly so? Explicit values of L(Modg) are known only when g = 1. In this case one is simply asking fo...

Compute L(Modg) explicitly for small g≥ 2. In principle L(Modg) can be computed for any given g. The point is that one can first bound the degree of L...

Is there a unique (up to conjugacy) minimal dilation pseudo-Anosov in Modg? Note that this is true for g = 1; the unique minimum is realized by the co...

(Shortest Teichm¨ uller loop in a stratum). For each fixed g≥ 2, and for each r-tuple as above, give upper and lower bounds for λg(k1,...,k r):= inf {...

Does spec(Ig(k)) have bounded multiplicity for k≥ 3? One way to get around unbounded multiplicities is to look at the simple length spectrum, which is...

(Simple length spectrum). Does the simple length spectrum of Mg, endowed with the Teichm¨ uller metric, have bounded multiplicity? If so, how does the...

Give an algorithm which tells whether or not any given pseudo-Anosov is represented by a simple closed Teichm¨ uller geodesic, and also whether or not...

Give upper and lower bounds for L(Ig(k)) for all k≥ 2 which are of the same order of magnitude. In [ FLM] bounds on L(H) are given for various special...

For various subgroups H <Modg, compute the density of spec(H) in spec(Modg) and the density of H∩Pg inPg. In particular, what is the density of spec(M...

( Topological Schottky Problem). Understand the homotopy type of Jc g and use it to compute H •(Jc g ) and H • c (Jc g ). The first interesting case i...

Determine whether or not H•(Tg; Z[1/2])− is always a finitely generated Z[1/2]- module. Does the infinite topology of Tg comes from Jg? To get one’s h...

Determine good bounds for the homological dimension (or the CW-dimension) ofT c g....

Try to understand the “topology at infinity” of T c g. In particular, try to compute Hk ∞(T c g ) for k in some range k≥do. Alternatively, try to comp...

Compute H • ∞(T c 3 ). The homology of T c g is related to that of Tg via the Gysin sequence. In order to apply it, one needs to understand the topolo...

Compute the Spg(Z)-moduleHk c (T c,red g ) in some range k≥ko. 3. Finiteness and Torelli spaces 71 We already know that H 6g−7 c (T c,red g ) = H0(Bc ...

Each component of Hc g is simply connected. This is trivially true in genus 2, where there is one component which is all of h2. If true in genus 3, it...

Investigate the topology of Hg andHc g and their components. Specifically, compute their homology and the cohomology at infinity of Hc g,α. The period...

Every subgroup of finite index in ModS contains a congruence subgroup. V. Voevodsky had indicated (in a personal communication) a beautiful applicatio...

s it true that any normal subgroup is commensurable with such a subgroup? Recall that two subgroups Γ 1, Γ2 of a group G are commensurable if the inte...

s it possible that all nontrivial (i.e., ̸= 1 ) elements of a normal subgroup of ModS are pseudo-Anosov? To the best of my knowledge, this question wa...

s there a constant NS, depending only on S, such that the following holds? Let f∈ ModS and let t =t±1 α1◦t±1 α2◦···◦ t±1 αn is a Dehn multi-twist. If ...

s it true that H 1(Γ) = 0 for any subgroup Γ of finite index in ModS? It is well known that H 1(ModS) = 0. In his 1999 MSU Ph.D. thesis F. Taherkhani ...

Does ModS has the Kazhdan property (T)? A positive answer would imply the positive answer to the previous question, but this problems seems to be much...

f f is pseudo-Anosov element of a mapping class group ModS with sufficiently big dilatation coefficient, then the subgroup of ModS normally generated by f...

Are there any other relations between N -th powers Tγ = tN γ of Dehn twists for sufficiently high N? In other words, do the above relations provide a pr...

s the subgroup of ModS generated by the N -th powers of all elements of ModS of infinite index in ModS for sufficiently big N? Notice that such a subgro...

LetS and R be closed surfaces. Let Γ be a subgroup of finite index in ModS. If the genus of R is less than the genus of S, then there is no homomorphi...

For every finitely generated subgroups G of ModS, the group Φf (G) is nilpotent. For a a little bit more detailed discussion, see [ I4], Section 10.10...

Is there an endomorphism of the mapping class group of a (closed) orientable surface onto an infinite index infinite subgroup? If there is one such en...

Let Γ be a subgroup of finite index in the mapping class group Mod1,2 and let φ: Γ → Γ be an automorphism. Is φ the restriction of an automorphism of ...

Let g > hand let Γ be a finite index subgroup of Modg. Assume that g≥ 3 and φ: Γ → Modh is a homomorphism. (a) Is the image of φ necessarily finite? (...

Let Γ be a finite index subgroup of Modg. Is it true that H 1(Γ; Z) = 0? There are some partial answers to Problem 2.6. If g≥ 3 and if Γ contains the ...

Suppose that ta1ta2··· tan = 1 in Modg, where n≥ 1. Let G denote the quotient H1(S)/⟨[a1], [a2],..., [an]⟩, where [ai] denotes the homology class of t...

Given a factorization ta1ta2··· tan = 1 of the identity into a product of right Dehn twists in Modg, is it always possible to lift this factorization ...

Suppose that a mapping class f ∈ Modb g, b≥ 1, is a product of right Dehn twists. Does there exist a constant Cf, depending on f, such that whenever f...

Compute ϕ(g,n ). Is it constant? If not, for g < h, compare ϕ(g,n ) and ϕ(h,n ). 90 M. Korkmaz...

Let g≥ 3 and b≥ 1. Let a1,a 2,... be an infinite sequence of nonseparating simple closed curves on an oriented surface S of genus g with b boundary co...

Let r be a positive integer and let Γr be the (normal) subgroup of Modg generated by the rth powers of all Dehn twists. Is Γr of infinite index? If we...

(a) It is known from [33] that H2(Mod3; Z) and H2(Mod1 3; Z) are either Z or Z⊕ Z2. What are they? Also compute H2(PModb 3,p; Z) for all p and b. (b) ...

(a) Is it possible to generate the mapping class group of a closed nonori- entable surface by two elements? (b) Is it possible to generate the mapping...

Compute the (outer) automorphism group of Mod(N )....

Letg >h, and let N andN ′ denote the closed nonorientable surfaces of genera g andh respectively. Is it true that any homomorphism φ: Mod(N )→ Mod(N ′...

Let φ: Mod( N )→ Mod(N ) be a homomorphism such that the image of φ is of finite index. Is φ necessarily an automorphism? How about if we take the dom...

Study homomorphisms Modg→ Mod(N ) and Mod(N )→ Modg. 92 M. Korkmaz It is known by the work of Birman and Chillingworth [ 3] that the mapping class gro...