Mathematics Problem Archive

(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajec...

(Renormalization of billiards in polygons). Is there a natural dynamical system acting on the space of billiards in polygons so as to allow a useful r...

(Characterization of Veech surfaces). Characterize all Veech surfaces (for each stratum of each genus). This problem is trivial in genus one; in genus...

(Fuchsian groups). Which Fuchsian groups are realized as Veech groups? Which subgroups of the mapping class group appear as Veech groups? This is equi...

(Purely cyclic). Is there a Veech group that is cyclic and generated by a single hy- perbolic element? Equivalently, is there a pseudo-Anosov map such...

(Algorithm for Veech groups). Is there an algorithm for determining the Veech group of a general translation surface or quadratic differential? An inte...

(Orbits of square-tiled surfaces). Classify the SL(2, R) orbits of square-tiled sur- faces in any stratum. Describe their Teichm¨ uller discs. A parti...

(Minimal sets). Describe the minimal sets for the SL(2, R)-action on H1(α) and Q1(β). Since Veech surfaces give rise to minimal sets, this problem gen...

(Kernel foliation). IsN a complex-analytic (complex-algebraic) orbifold? When is dimCN = dim CO +n−m? On the other hand when does N coincide with the ...

(Decomposition of surfaces). Given a connected component of the stratum H(α) of Abelian differentials (or of quadratic differentials Q(β) find those con...

(Lyapunov exponents). Study individual Lyapunov exponents of the Teichm¨ uller geodesic flow: – for all known SL(2; R)-invariant subvarieties; – for s...

(Dynamical Hodge decomposition). Study properties of distributions of the La- grangian subspaces in H 1(S; R) defined by the Teichm¨ uller geodesic fl...

(Converse to dichotomy). Characterize translation surfaces for which (1) the set of minimal directions coincides with the set of uniquely ergodic dire...

(Quadratic asymptotics for any surface). Is it true that every translation surface or quadratic differential has exact quadratic asymptotics for the nu...

(Error term for counting functions). What can be said about the error term in the quadratic asymptotics for counting functions N ((X,ω ),L )∼c·L2 on a...

(Topology of strata). Is it true that the connected components of the strata H(α) and of the strata Q(β) areK(π, 1)-spaces (i.e. their universal cover...

(Exceptional Strata). Find a geometric invariant which distinguishes differ- ent connected components of the four exceptional strata Q(−1, 9),Q(−1, 3, ...

Given a subgroup G <MCG (S), how is the geometry of ΓG related to the dynamics of the action of G on Thurston ’s compactification of Teichm¨ uller spa...

Does the converse hold in the above theorem without the assumption that G is free? The gist of Problem 2.2 is to find an extension of the Bestvina-Fei...

Does there exist an algorithm which produces the integer M in Theorem 3.1, given φ1,...,φ n?...

If H⊂MCG (S) is finite rank free subgroup whose nonidentity elements are pseudo-Anosov, is H a Schottky group? For specific examples on which to test ...

Is every finite rank subgroup of Whittlesey’s group a Schottky subgroup of MCG (S)? As a consequence of Theorem 2.1, if H <MCG (S) has a finite index ...

Give examples and constructions of virtual Schottky subgroups of MCG (S). One such construction is due to Honglin Min, currently a doctoral candidate ...

Do there exist two surfaces S,S ′, closed and of genus ≥ 2, such that MCG (S) contains a subgroup isomorphic to π1(S′) all of whose nontrivial element...

Do there exist two surfaces S,S ′, closed and of genus ≥ 2, and a subgroup G <MCG (S) isomorphic to π1(S′), so that ΓG is word hyperbolic? Or so that ...

Does there exist any non-virtually free, finitely generated subgroup G< MCG (S) whose nontorsion elements are all pseudo-Anosov? Does G exist so that ...

Does there exist a simple cycle of dihedral subgroups of MCG (S) so that the associated reflection group P injects in MCG (S)? So that the image of P ...

Suppose that G <MCG (S) is a finite co-area Veech subgroup. What can one say about ΓC? In particular, does it contain ΓG with finite index?...

Explore ΓC for other free subgroups G< MCG (S), for example free subgroups generated by high powers of Dehn twists about a pair of filling curves. IfG...

Are the Leininger–Reid subgroups geometrically finite, with cusp groups the reducible cyclic subgroups?...

If G< MCG (S) is geometrically finite with cusp groups H1,...,H n, what can be said about the geometric properties of the group ΓG? Does it have usefu...

Forg≥ 2, does Γg contain a purely pseudo-Anosov surface subgroup? The paper is organized as follows. In §2, we discuss the existence of surface subgro...

Are there are only finitely many Γg-conjugacy classes of purely pseudo-Anosov surface subgroups of any fixed genus? Of course given that Question 1.1 ...

LetG∼=π1(S2g)→ Γg be the injection given by Theorem 3.1. Consider ∂∞(G) which can be canonically identified with the circle at infinity of the univers...

Is Γg GFERF for g≥ 2?...

LetH be a convex cocompact subgroup of Γg. Is Γg H-separable? Just focusing on surface subgroups, we can ask:...

LetH be a surface subgroup of Γg. Is Γg H-separable? For recent progress on various classes of subgroups of Γ g that are separable, we refer the reade...

Does there exist a closed hyperbolic 4-manifold X that is the total space of a smooth fiber bundle Σg→X→ Σh? We will call such an X a surface bundle o...

Does there exist a closed hyperbolic 4-manifold that is a surface bundle over a surface where the genus of the fiber and base is 2? 4.2. Some evidence...

Let M be a closed hyperbolic 4-manifold. Then all the Seiberg-Witten invariants of M vanish. The relevance of this is given in the following propositi...

Does there exist a closed hyperbolic 4-manifold X for which no finite cover admits a symplectic structure? (ie X is not virtually symplectic.) We note...

Forg,h≥ 2, does there exist a short exact sequence: 1→π1(Σg)→ Γ→π1(Σh)→ 1 for which Γ is a word hyperbolic group? Arguing as in the proof of Theorem 4...

Does there exist a cocompact Fuchsian subgroup of ∆5 that misses the com- pactification locus? Given such a Fuchsian subgroup F <∆5, Agol produces a p...

Let Γ be a lattice in SO(m, 1), m≥ 3 or SU(q, 1), q≥ 2 which is admissable for Γg. Does Γ inject in Γg? Can there be purely pseudo-Anosov representati...

Which 1-ended admissable word hyperbolic groups G inject in Γg (as purely pseudo-Anosov subgroups)? If such an injection exists does there exist a con...

If R = Q(q1,q 2), is the above map from Bn toHn(q1,q 2) injective?...

What are the equivalence classes of braids modulo the moves • ab↔ba, and • b↔σnι(b)? In other words, what happens if the Markov move b↔σ−1 n ι(b) is o...

What can be said about the dimensions of Dλ?...

How much of this paper can be generalized to the Birman-Wenzl-Murakami algebra? In this direction, John Enyang [ Eny04] has shown that the Birman-Mura...

Is there a homological definition of representations of the Birman-Wenzl- Murakami algebra? I believe the answer to this is yes. Furthermore, the homo...