Mathematics Problem Archive
Problem 5: — Develop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces.
v1.3 research notesDevelop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces. 14. Geometric properties of the mapping...
Problem 6: — For a closed surface of genus g≥ 3, is the Torelli subgroup of Mg,m undistorted?
v1.3 research notesFor a closed surface of genus g≥ 3, is the Torelli subgroup of Mg,m undistorted? More generally, find a distorted finitely generated subgroup of Mg,m....
Problem 7: — Is the mapping class group linear?
v1.3 research notesIs the mapping class group linear? A locally compact group Γ is said to satisfy the Haagerup approximation property or is a-T- menable if there exists...
Problem 8: — Is the mapping class group a-T-menable?
v1.3 research notesIs the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245...
Problem 1 — (Geodesics on general flat surfaces).
v1.3 research notes(Geodesics on general flat surfaces). Describe the behavior of geodesics on general flat surfaces. Prove (or disprove) the conjecture that the geodesi...
Problem 2 — (Billiards in general polygons).
v1.3 research notes(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajec...
Problem 3 — (Renormalization of billiards in polygons).
v1.3 research notes(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...
Problem 4 — (Characterization of Veech surfaces).
v1.3 research notes(Characterization of Veech surfaces). Characterize all Veech surfaces (for each stratum of each genus). This problem is trivial in genus one; in genus...
Problem 5 — (Fuchsian groups).
v1.3 research notes(Fuchsian groups). Which Fuchsian groups are realized as Veech groups? Which subgroups of the mapping class group appear as Veech groups? This is equi...
Problem 6 — (Purely cyclic).
v1.3 research notes(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...
Problem 7 — (Algorithm for Veech groups).
v1.3 research notes(Algorithm for Veech groups). Is there an algorithm for determining the Veech group of a general translation surface or quadratic differential? An inte...
Problem 8 — (Orbits of square-tiled surfaces).
v1.3 research notes(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...
Problem 9 — (Orbit closures for moduli spaces).
v1.3 research notes(Orbit closures for moduli spaces). Determine the closures of the orbits for the GL+(2, R)-action on H(α) andQ(β). Are these closures always complex-a...
Problem 10 — (Ergodic measures).
v1.3 research notes(Ergodic measures). Classify the ergodic measures for the action of SL(2, R) on H1(α) andQ1(β). McMullen [ McM3] has solved Problems 9 and 10 in the c...
Problem 11 — (Minimal sets).
v1.3 research notes(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...
Problem 12 — (Analog of Ratner theorem).
v1.3 research notes(Analog of Ratner theorem)....
Problem 13 — (Kernel foliation).
v1.3 research notes(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 ...
Problem 14 — (Decomposition of surfaces).
v1.3 research notes(Decomposition of surfaces). Given a connected component of the stratum H(α) of Abelian differentials (or of quadratic differentials Q(β) find those con...
Problem 15 — (Lyapunov exponents).
v1.3 research notes(Lyapunov exponents). Study individual Lyapunov exponents of the Teichm¨ uller geodesic flow: – for all known SL(2; R)-invariant subvarieties; – for s...
Problem 16 — (Dynamical Hodge decomposition).
v1.3 research notes(Dynamical Hodge decomposition). Study properties of distributions of the La- grangian subspaces in H 1(S; R) defined by the Teichm¨ uller geodesic fl...
Problem 17 — (Converse to dichotomy).
v1.3 research notes(Converse to dichotomy). Characterize translation surfaces for which (1) the set of minimal directions coincides with the set of uniquely ergodic dire...
Problem 18 — (Quadratic asymptotics for any surface).
v1.3 research notes(Quadratic asymptotics for any surface). Is it true that every translation surface or quadratic differential has exact quadratic asymptotics for the nu...
Problem 19 — (Error term for counting functions).
v1.3 research notes(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...
Problem 20 — (Topology of strata).
v1.3 research notes(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...
Problem 21 — (Exceptional Strata).
v1.3 research notes(Exceptional Strata). Find a geometric invariant which distinguishes differ- ent connected components of the four exceptional strata Q(−1, 9),Q(−1, 3, ...
Problem 1.1 — Given a subgroup G <MCG (S), how is the geometry of ΓG related to the dynamics of the action of G on Thurston ’s comp…
v1.3 research notesGiven 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...
Problem 2.2 — Does the converse hold in the above theorem without the assumption that G is free?
v1.3 research notesDoes 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...
Problem 3.3 — Does there exist an algorithm which produces the integer M in Theorem 3.1, given φ1,...,φ n?
v1.3 research notesDoes there exist an algorithm which produces the integer M in Theorem 3.1, given φ1,...,φ n?...
Problem 3.4 — If H⊂MCG (S) is finite rank free subgroup whose nonidentity elements are pseudo-Anosov, is H a Schottky group?
v1.3 research notesIf 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 ...
Problem 3.5 — Is every finite rank subgroup of Whittlesey’s group a Schottky subgroup of MCG (S)?
v1.3 research notesIs 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 ...
Problem 3.6 — Give examples and constructions of virtual Schottky subgroups of MCG (S).
v1.3 research notesGive examples and constructions of virtual Schottky subgroups of MCG (S). One such construction is due to Honglin Min, currently a doctoral candidate ...
Problem 4.1 — Do there exist two surfaces S,S ′, closed and of genus ≥ 2, such that MCG (S) contains a subgroup isomorphic to π1(S′…
v1.3 research notesDo 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...
Problem 4.2 — Do there exist two surfaces S,S ′, closed and of genus ≥ 2, and a subgroup G <MCG (S) isomorphic to π1(S′), so that Γ…
v1.3 research notesDo 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 ...
Problem 4.3 — Does there exist any non-virtually free, finitely generated subgroup G< MCG (S) whose nontorsion elements are all pse…
v1.3 research notesDoes there exist any non-virtually free, finitely generated subgroup G< MCG (S) whose nontorsion elements are all pseudo-Anosov? Does G exist so that ...
Problem 4.4 — Does there exist a simple cycle of dihedral subgroups of MCG (S) so that the associated reflection group P injects in…
v1.3 research notesDoes 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 ...
Problem 5.4 — Suppose that G <MCG (S) is a finite co-area Veech subgroup.
v1.3 research notesSuppose 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?...
Problem 5.5 — Explore ΓC for other free subgroups G< MCG (S), for example free subgroups generated by high powers of Dehn twists ab…
v1.3 research notesExplore 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...
Problem 6.1 — Are the Leininger–Reid subgroups geometrically finite, with cusp groups the reducible cyclic subgroups?
v1.3 research notesAre the Leininger–Reid subgroups geometrically finite, with cusp groups the reducible cyclic subgroups?...
Problem 6.2 — If G< MCG (S) is geometrically finite with cusp groups H1,...,H n, what can be said about the geometric properties of…
v1.3 research notesIf 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...
Question 1.1 — Forg≥ 2, does Γg contain a purely pseudo-Anosov surface subgroup?
v1.3 research notesForg≥ 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...
Question 2.1 — Let M = H3/Γ be a finite volume hyperbolic 3-manifold, does Γ contain a surface subgroup.
v1.3 research notesLet M = H3/Γ be a finite volume hyperbolic 3-manifold, does Γ contain a surface subgroup. This was answered in [ 8] for non-compact but finite volume ...
Question 3.2 — Are there are only finitely many Γg-conjugacy classes of purely pseudo-Anosov surface subgroups of any fixed genus?
v1.3 research notesAre 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 ...
Question 3.3 — LetG∼=π1(S2g)→ Γg be the injection given by Theorem 3.1.
v1.3 research notesLetG∼=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...
Question 3.4 — Is Γg GFERF for g≥ 2?
v1.3 research notesIs Γg GFERF for g≥ 2?...
Question 3.5 — LetH be a convex cocompact subgroup of Γg.
v1.3 research notesLetH be a convex cocompact subgroup of Γg. Is Γg H-separable? Just focusing on surface subgroups, we can ask:...
Question 3.6 — LetH be a surface subgroup of Γg.
v1.3 research notesLetH 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...
Question 4.1 — Does there exist a closed hyperbolic 4-manifold X that is the total space of a smooth fiber bundle Σg→X→ Σh?
v1.3 research notesDoes 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...
Question 4.3 — Does there exist a closed hyperbolic 4-manifold that is a surface bundle over a surface where the genus of the fiber…
v1.3 research notesDoes 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...
Conjecture 4.4 — Let M be a closed hyperbolic 4-manifold.
v1.3 research notesLet 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...
Question 4.6 — Does there exist a closed hyperbolic 4-manifold X for which no finite cover admits a symplectic structure?
v1.3 research notesDoes 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...