Mathematics Problem Archive

Is there a classification of the strongly irreducible Heegaard splittings of a given $3$-manifold?...

Do higher-dimensional smooth manifolds always admit multisections? What is the correct generalization of uniqueness up to stabilization for multisecti...

A band joining the components of a two-component link $L\subset S^3$ is called complicated if either its core cannot be isotoped to meet a splitting s...

Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function. Classify its rational critical points. Deduce properties of the rational cohomo...

Determine the finiteness properties of Ig. For which k is Hk(Ig) finitely gen- erated? For which k is there a K(Ig, 1) with finite k-skeleton (one say...

Letγ1,···,γ 2g be the standard basis of Z2g. Study the function L = ∑ Lγi:Yg→ [0,∞) as a Morse function. Find critical sets and deduce properties of t...

Work out the details of this construction of the completion Yg of Yg. Show that L:Yg→ [0,∞) extends to L:Yg→ [0,∞) and is a proper map. Ideally, inclu...

Find a (6g− 8)-obstructor complex L and a proper expanding map F:L× [0,∞)→T ≥ϵ g 3For concreteness we triangulate each sphere as the join of 0-spheres...

Show that there are many quasihomomorphisms f:MCG (Sg)→ R that satisfy (1) and (2) above plus (3) f is bounded on every G(q). I remark that the conseq...

(Ends spectrum). What are the possibile values of ends(Modg,H ) for finitely- generated subgroups H <Modg? It is well-known that the moduli space Mg h...

Compute CommModg (Γ) for various subgroups Γ< Modg. Paris-Rolfsen and Paris (see, e.g., [ Pa]) have proven that most subgroups of Mod g stabiliz- ing ...

(Volume spectrum). Determine for each 1≤ k≤ 3g− 3 the image of Volk: Xg(Γ)→ R. Determine the union of all such images as Γ ranges over all finitely pr...

Does there exist some Modg,g ≥ 2 that contains a subgroup Γ isomorphic to a cocompact (resp. noncocompact) lattice in SO(m, 1) with m≥ 5 (resp. m≥ 4)?...

(Holomorphic representatives). Find an algorithm or a group-theoretic invariant which determines or detects whether or not a given representation ρ: π...

(Normal subgroups). Let Γ be a finitely generated normal subgroup of Modg, whereg≥ 3. Must Γ be commensurable with Modg or with Ig? One way of constru...

Is it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free? Gromov discovered the analogous phe...

Forg≥ 2, determine the irreducible factors of the graded pieces of the Malcev Lie algebra tg ofIg as Sp-modules. While Hain gives in [ Ha3] an explici...

Give a proof of Theorem 2.13 which does not depend on the classification of finite simple groups. 22 B. Farb To complete the picture, one would like t...

(Frequency of low symmetry). Let H denote the set of integers g≥ 2 such that N (g) = 8( g + 1). Find the s0 for which the series ∑ g∈Hg−s converges ab...

(Automorphism groups with special properties). LetP be a property of finite groups, for example being nilpotent, solvable, or a p-group. Prove a versi...

(Nonarithmetic extremal surfaces). Give answers to all of the above problems on automorphisms of Riemann surfaces for the collection of non-arithmetic...

(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...

(Density of pseudo-Anosovs). LetP denote the set of pseudo-Anosov ele- ments of Modg. Then d(P) = 1. J. Maher [ Mah] has recently proven that a random...

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...

(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...