Mathematics Problem Archive

Consider the Artin group A5 (the braid group on six strings) divided by the relation (a1a2a3a4)5 = a5a4a3a2a2 1a2a3a4a5. Is it true that every positiv...

Is the Hurwitz problem for mapping class group factorizations decidable? Are there interesting criteria which can be used to conclude that two given f...

(Donaldson). Is it possible to enumerate all matching paths in a Lefschetz fibration with given monodromy factorization?...

(Smith). Is there an a priori upper bound on the length of any factorization of the boundary twist δ as a product of positive Dehn twists in Mapg,n? E...

For which T∈ Map+ g,n is it possible to classify factorizations of T as a product of positive Dehn twists in Mapg,n? In particular, for which T is the...

Given two factorizations of the boundary twist δ as a product of positive Dehn twists along nonseparating curves in Mapg,n, such that the total spaces...

The ideas which were just described relate to the beginning of the lower central series of Ig. There is also the lower central series of Kg. The corre...

Assume, for this problem, that M is a 3-manifold with non-empty boundary. Then, on one side of the double coset HφH the handlebody subgroup needs to b...

How is the Nielsen-Thurston trichotomy related to the question of whether the distance is 0, 1, 2 or≥ 3? The next 3 problems concern the very non-cons...

Find an algorithm to compute the distance d(φ) of an arbitrary element φ∈M. We note that an algorithm to compute shortest paths between fixed vertices...

Knowing that d(φ)≤ 1, can we decide whether d(φ) = 0? Geometrically, if a Heegaard splitting is weakly reducible, can you decide if it’s reducible?...

Knowing that d(φ)≥ 1, can we decide whether it is ≥ 2? Knowing that it’s ≥ 2, can we decide whether it is ≥ 3?...

Schleimer has proved in [42] that each fixed 3-manifold M has a bound on the distances of its Heegaard splittings. Study this bound, with the goal of ...

Study the handlebody subgroup of Mg. A simplified presentation which would reveal new things about its structure, and/or anything new about its coset ...

Recall that we noted, earlier, that every genus g Heegaard splitting of every homology 3-sphere is obtained by allowing ϕ to range over Ig. We also no...

Uncover the structure in the mapping class group that relates to the classifica- tion theorem for the Heegaard splittings of graph manifolds in [44]....

Given a normal subgroup Gg ofMg, what basic properties are needed in a complex G(Sg) of curves on Sg so that Mg will turn out to be naturally isomorph...

Hempel’s distance function was chosen so that it would capture the geometry, and indeed it does that very well, yet in some ways it feels unnatural. T...

This one is a warm-up. Given α∈M g, say as a product of Dehn twists, expressβ∈M 2g+1 as a related product of Dehn twists. With that in hand, observe t...

In [43] it is proved that in the case of the trivial genus g surface bundle, i.e. Sg×S1 the bundle-related splitting is unique, up to equivalence. Are...

A 3-manifold is fibered if it admits a surface bundle structure. It is virtually fibered if it has a finite-sheeted cover that admits a surface bundle...

Is there a natural quotient complex of any one of the complexes discussed in §1 which might be useful for the construction of non-faithful representat...

Study the double coset HφH inM, using new finite or infinite quotients of M. In this regard we stress finite, because a principle difficulty when this p...

Are there quotients of Ig orKg in which the intersection of either Ig orKg with the handlebody group Hg is sufficiently tractible to allow one to study ...

Develop techniques to describe the sets Cχ,θ....

Show that the inclusion of Proposition 5 is a bijection. This will probably require more thought about the analytical and geometric constructions whic...

Given a topological description of fk0,fk1 describe fk0+k1. A good understanding of this would enable one to drop the rather artificial introduction o...

Reproduce the classification of manifolds with ω.K (ω)< 0 by studying the sets Cχ,θ for θ< 0. There is a network of interesting questions dealing with...

Analyse the monodromy of Lefschetz fibrations on manifolds with K(ω) = 0. Related to this is the general question of understanding the place of comple...

Find special features of the monodromy of algebraic surfaces. There is some good motivation for this coming from at least three directions • The probl...

Investigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩....

Let Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G. Then the symplectic structures ωB generate t...

Suppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G )/G. Then D is a scalar multiple ...

Decompose the representation on H0 into irreducible representations of ModΣ. When G = U(1), and Σ is the 2-torus, Hom( π,G )/G naturally identifies wi...

Find a similar geometric interpretation for the irreducible constituents for compact nonabelian groups G....

If r≥ 3, the action of Out(π) on Hom(π,G ) is ergodic. Using calculations in [ 43], this conjecture has been proved [ 47] when all of the simple facto...

Determine necessary and sufficient conditions on a general representation ρ for its orbit ModΣ· [ρ] to be dense. The case when G = SU(2) and Σ an n-hole...

Construct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-character variety....

Suppose that b = 0 (Σ is closed). For each integer 1≤k≤ 2g +b− 2, the ModΣ-action on the component e−1(2− 2g +b +k) of Hom(π,G ) is ergodic. When b = ...

Determine the smallest dimensional manifold homotopy-equivalent to Σ upon which ModΣ acts compatibly with the outer action of ModΣ on π1(Σ)....

Find general conditions which ensure that (10) is proper. The level set R3∩κ−1(2) consists of characters of abelian representations, and ModΣ is ergod...

Determine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2. The level sets for t> 6 contains wandering domains...

Find a point ρ∈ Hom(π, SL(2, C)) such that the closure of its orbit ModΣ· [ρ] meets both the image of the unitary characters Hom(π, SU(2)) and the clo...

Find a substitute for convex cocompactness in higher rank which includes the above examples of proper ModΣ-actions, and for which Eρ is proper. The wo...

If k = 1, then U is onto. In general a PSL(2, R)-representation with dense image lies in Image(U). 220 W. Goldman...

For a fixed number R > 0, is there is compact subset K(R) of moduli space containing the projection of every Teichm¨ uller geodesic γ which satisfies ...

Describe the space of geodesic currents for Mg,m. Is the set of weighted sums of Dirac masses at the pairs of fixed points of pseudo-Anosov elements d...

Does the above definition of a convex cocompact subgroup of Mg,m coincide with the definition of Farb and Mosher in [FMo]? Is the natural extension of...

Develop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces. 14. Geometric properties of the mapping...

Is the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245...