Mathematics Problem Archive

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...

DoesX3 equal 0 in Zn? Presumably some extra relations should be added to Zn, such as σ1X2 = tX2, or something more general....

What extra relations should be added to Zn to make it finite-dimensional?...

How much of this paper can be generalized to Zn? It might be easier to first study these questions for the quotient of Zn by the relation X4 = 0....

Does the Teichm¨ uller space for Sg admit an equivariant deformation retraction onto a cocompact spine whose dimension is equal to 4g− 5, the virtual ...

Develop a metric theory of Outer space. The elements of infinite order in GL( n, Z) that are diagonalizable over C act as loxodromic isometries of X. ...

Describe the geometry of the axis bundle (and associated objects) for an iwip acting on Outer Space. 322 M. Bridson and K. Vogtmann...

Do mapping class groups or Out(Fn) satisfy the Baum-Connes conjecture? Does Out(Fn) satisfy the Novikov conjecture? An approach to proving these conje...

Does there exist a compactification of the spine of Outer space satisfying Rosen- thal’s conditions? Same question for the complex of arc systems fill...

Can one construct a cocompact EG with dimension equal to the virtual coho- mological dimension of the mapping class group of a closed surface?...

For n > 3, does Aut(Fn) have a subgroup of finite index with positive first betti number? Another finite-index subgroup of Aut( F3) mapping onto Z was...

If there is a homomorphism from a subgroup of finite index in Aut(Fn) onto a subgroup of finite index in GL(m, Z), then must m≥n− 1? 324 M. Bridson an...

If m<n − 1 and H⊂Aut(Fn) is a subgroup of finite index, then does every homomorphism H→ GL(m, Z) have finite image? Similar questions are interesting ...

Forn≥ 4, do subgroups of finite index in Aut(Fn) have Property F A? A promising approach to this last question breaks down because we do not know the ...

Fix a basis for Fn and let An−1⊂ Aut(Fn) be the copy of Aut(Fn−1) corre- sponding to the first n− 1 basis elements. Let φ:Aut(Fn)→G be a homomorphism ...

What is the least integer δ such that Out(Fn) acts without a global fixed point on a complete CAT (0) space of dimension δ? And what is the least dime...

If n ≥ 4, then can Out(Fn) act without a global fixed point on a finite- dimensional CAT(0) cube complex?...

Does Out(F3) have a faithful representation into GL(m, C) for some m∈ N? Note that braid groups are linear [ 8] but it is unknown if mapping class gro...

If n≥ 4 and g≥ 1, does every homomorphism from Aut(Fn) to Mod±(Sg) have finite image? By [ 21], one cannot obtain homomorphisms with infinite image un...

Let Γ be an irreducible lattice in a semisimple Lie group of R-rank at least 2. Does every homomorphism from Γ to Out(Fn) have finite image? This is k...

Is there a theory of random walks on Outer space similar to that of Kaimanovich and Masur for Teichm¨ uller space? Perhaps the most promising approach...

If a subgroup G⊂Out(Fn) is not virtually abelian, then is H 2 b (G; R) infinite dimensional? Ifm≥n then there are obvious embeddings GL( n, Z)→ GL(m, ...

For which values of m does Out(Fn) embed in Out(Fm)? What is the minimal such m, and is it true for all sufficiently large m? It has been shown that whe...

Is there a map Out(Fn)→ Out(Fm) that induces an isomorphism on homology in the stable range? A number of the questions in this section and (2.2) ask w...

For which values of n and m is Q(n,m ) infinite? Is Q(3, 5) infinite?...