Mathematics Problem Archive

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?...

Forn> 3, does Aut(Fn) have property (T)? The corresponding question for mapping class groups is also open. If Aut( Fn) were to have Property (T), then...

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?...

Can Q(n,m ) have infinitely many finite quotients? Is it residually finite?...

Can one detect the growth of a surface or free-group homomorphism by its action on the homology of a characteristic subgroup of finite index? Notice t...

Classify those φ∈ Aut(Fn) for which Fn ⋊φ Z is automatic and those for which it is CAT(0). Of central importance in trying to understand mapping tori ...

Is there an alogrithm to decide isomorphism among groups of the form F ⋊ Z. In the purest form of this question one is given the groups as finite pres...

Is the conjugacy problem solvable in Out(Fn)? Martin Lustig posted a detailed outline of a solution to this problem on his web page some years ago [ 6...

Where precisely does the rational homology of Aut(Fn) stabilize? And for Out(Fn)? There are only two known non-trivial classes in the (unstable) ratio...

Are Morita’s original cycles non-trivial in homology? Are the generalizations due to Morita and to Conant and Vogtmann non-trivial in homology? No oth...

Do the Morita classes generate all of the rational homology of Out(Fn)? The maximum dimension of a Morita class is about 4 n/3. Morita’s cycles lift n...

Is the image of the second Morita class in H8(GL(6, Z); Q)) non-trivial? For further discussion of the cohomology of Aut( Fn) and Out( Fn) we refer to...

Is there a set of simple Steinberg-type relations for the mapping class group? There is also a presentation of Aut( Fn) coming from the action of Aut(...

Can Out(Fn) and Mod±(Sg) be obtained as a pushout of a finite subsystem of their finite subgroups, i.e. is either the fundamental group of a developab...

Establish finiteness properties of the kernel IA(n) of the map from Out(Fn) to GL(n, Z). In particular, determine whether IA(n) is finitely presentabl...

What are the Dehn functions of Aut(Fn) and Out(Fn) for n> 3?...

What are the higher-dimensional isoperimetric functions of GL(n, Z), Aut(Fn)and Out(Fn)?...

Is Aut(Fn) automatic for n> 3?...

The natural homomorphisms ( Λ∗Λ3HQ )Sp →H ∗(Mg,∗; Q), (Λ∗UQ)Sp→H ∗(Mg; Q) induce isomorphisms ( Λ∗Λ3H ∗ Q/ ( [12]torelli⊕ [22] ))Sp ∼=R∗(Mg,∗) ( Λ∗U ∗...