Mathematics Problem Archive

(Canonical basepoints for Mg). Find other properties of automorphisms or automorphism groups that determine a unique point of Mg. For example, is ther...

(Number of Hurwitz surfaces). Give a formula for the number of Hurwitz surfaces of genus g. What is the frequency of those g for which there is a uniq...

(Conjugator length bounds). Prove that there exist constants C,K, depend- ing only on S, so that if u,v ∈ Modg are conjugate, then there exists g ∈ Mo...

(Fast conjugacy problem). Find a polynomial time algorithm to solve the con- jugacy problem in Modg. Is there a quadratic time algorithm, as for the w...

(Almost convexity). Does there exist a finite generating set for Modg for which it is almost convex? One would also like to know the answer to this qu...

Is Teich(Σg), endowed with the Teichm¨ uller metric, almost convex? Note that Cannon proves in [ Ca] that fundamental groups of closed, negatively cur...

(Generalized word problem). Determine the subgroups H in Modg for which the generalized word problem is solvable. Give efficient algorithms to solve the...

(Distortion). Find the possible distortions of subgroups in Modg. In particular, compute the distortions of Ig. Determine the asymptotics of the disto...

Determine which subgroups of Modg are quasiconvex with respect to some collection of geodesics. This question is closely related to, but different than...

Does every finitely presented subgroup H < Modg have solvable conjugacy problem? is it combable? automatic? Note that every finitely-generated subgrou...

Find a finitely presented subgroup H < Modg for which there are infinitely many conjugacy classes of finite subgroups in H. The motivation for this pr...

(Isomorphism problem for subgroups). Is the isomorphism problem for the collection of finitely presented subgroups of Modg solvable? Note that the iso...

Is there an algorithm to decide whether or not a given subgroup H <Modg is freely indecomposable? Whether or not H splits over Z? 28 B. Farb...

(Rational growth). Does Modg have rational growth function with respect to some set of generators? with respect to every set of generators? Of course ...

(Rational growth for properties). For which properties P is the function fP is rational? Densities. For any subset S⊂ Modg, it is natural to ask how c...

d(Ig) = 0. Even better would be to determine dlog(Ig). Conjecture 3.16 would imply that d(Ig(m)) = 0 for each m≥ 2. It is not hard to see that Ig(m) h...

(Logarithmic densities of the Johnson filtration). Determine the asymptotics of dlog(Ig(m)) both as g→∞ and as m→∞. Indeed, as far as I know, even the...

Give explicit upper and lower bounds for ent(Modg). Compute the asymptotics of ent(Modg) and of ent(Ig) as g→∞. Similarly for ent(Ig(k)) as k→∞....

(Inhomogeneity of all metrics). Let Teichg denote the Teichm¨ uller space of closed, genus g≥ 2 Riemann surfaces. Let h be any Riemannian metric (or a...

( Mg is maximal). For g ≥ 3 the smooth orbifold Mg does not finitely orbifold-cover any other smooth orbifold. A much stronger statement, which may be...

LetY be any finite cover of Mg, and let f:Y →Y be a finite order homeo- morphism. If f is homotopic to the identity, must f equal the identity?...

(Nonpositive curvature). For g≥ 2 the orbifold Mg admits no complete, finite volume Riemannian metric with nonpositive sectional curvatures uniformly ...

LetS be any surface with d(S)≥ 1. Then M does not admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the quas...

( Q-rank of moduli space). Cone(Mg) is homeomorphic to the (open) cone on the quotient Cg/ Modg 6. One can pose a stronger version of Conjecture 4.7 t...

(Mod g is Kahler). Forg≥ 3, the group Modg is a Kahler group, i.e. it is isomorphic to the fundamental group of a compact Kahler manifold. It was show...

(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...