Mathematics Problem Archive

(Algorithmic Schottky problem). Give an algorithm, in the sense of complexity theory over R, which takes as input a 2g×2g symplectic matrix representi...

(Coarse Schottky problem). Describe, as a subset of a g-dimensional Euclidean sector, the subset of Cone(Ag) determined by the Schottky locus in Ag. P...

(Distortion of the Schottky locus). Compute the distortion of the Schottky locus in Ag. 36 B. Farb A naive guess might be that it is exponential....

(Morita). Is H1(Kg, Z) finitely generated for g≥ 3? Note that Birman-Craggs-Johnson (see, e.g., [ BC, Jo1 ]) and Morita [ Mo2] have found large abelia...

(Interpolations). Letg≥ 3. For each subgroup L< ∧3H/H, determine whether or not π−1(L) is finitely generated. As for subgroups deeper down than Kg =Ig...

For each k≥ 1, the group (Ig)k is not finitely generated. Another test of our understanding of the Johnson filtration is the following....

Find H1(Ig(k), Z) for all k≥ 2. Generating sets for Ig. One difficulty in working with Ig is the complexity of its generating sets: any such set must ha...

(Cubic genset problem). Find a generating set for Ig withO(gd) many elements for some d≥ 3. Optimally one would like d = 3. In fact in §5 of [ Jo2], J...

(Sharp bounds for involution generating sets). For each g≥ 2, prove sharp bounds for the minimal number of involutions required to generate Modg. In p...

(Cohomological Dimension). Compute the cohomological dimension of Ig and ofKg. More generally, compute the cohomological dimension of Ig(k) for all k≥...

(Torelli finiteness). Determine the maximal number f (g) for which there is a K(Ig, 1) space with finitely many cells in dimensions ≤f (g). Here is wh...

Ig is finitely presented for g≥ 4. One thing we do know is that, in contrast to Mod g, neither Ig norKg has a classifying space which is homotopy equi...

Extend Akita’s result to 2<g < 7. Since Akita’s proof produces no explicit homology classes, the following seems fundamental....

(Explicit cycles). Explicitly construct infinitely many linearly independent cy- cles in H∗(Ig, Q) and H∗(Kg, Q). So, we are still at the stage of try...

Determine the subalgebras of H ∗(Ig,K ), for K = Q and K = F2, generated by H 1(Ig,K ). Note that H ∗(Ig,K ) is a module over Sp(2 g,K ). When K = Q t...

For which k≥ 1 is it true that Aut(Ig(k)) = Mod ± g? that Comm(Ig(k)) = Mod± g? Theorem 5.17 answers the question for k = 1, 2. It would be remarkable...

Give an elementary, purely combinatorial-topological and group-theoretic, proof of Hain ’s theorem. It seems that a solution to Problem 5.19 will like...

(Hain for Aut( Fn)). Give an explicit finite presentation for the Malcev Lie AlgebraL(IAn), where IAn is the group of automorphisms of the free group ...

(Malcev mod 2). Give an explicit finite presentation for the F2-Lie algebra L2(Ig,1). We can also build a Lie algebra using the Johnson filtration. Le...

(Lie algebra for the Johnson filtration). Is hg a finitely presented Lie algebra? If so, give an explicit finite presentation for it....

Compute H1(Modg[L]; Z). McCarthy and (independently) Hain proved that H1(Modg[L], Z) is finite for g≥ 3; see, e.g. Proposition 5.2 of [ Ha2]7. As disc...

(Picard number one conjecture for level L structures). Prove that H2(Modg[L]; Q) = Q when g≥ 3. More generally, compute H2(Modg[L]; Z) for all g≥ 3,L≥...

(Presentation for level L structures). Give an explicit finite presentation for Modg[L]. Once one has such a presentation, it seems likely that it wou...

(Actions on buildings). Determine all isometric actions ψ: Mod g→ Isom(Xn), whereXn is an n-dimensional Euclidean building, and n is sufficiently small ...

(Rigidity of the Mod g,1 action on S1). Is any faithful action ρ: Mod g,1→ Homeo+(S1) conjugate in Homeo+(S1) to the standard action, given in (19)? W...

(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...