Mathematics Problem Archive

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?...

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 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 ∗...

Prove (or disprove) that the even Mumford-Morita-Miller classes e2i∈H 4i(Ig; Q) are non-trivial, in a suitable stable range, as cohomology classes of ...

Let ug denote the graded Lie algebra associated to the prounipotent radical of the relative Malcev completion of Ig defined by Hain [29] and let ug→hQ...

Prove that all the secondary classes d2,d 3,··· are non-trivial. Here is a problem concerning the first class d1. Let C be a separating simple closed ...

Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg. In particular, determine whether the Magnus representation Ig,1→GL(2g; Z[H]) of t...

The classes µi are non-trivial for all i = 1, 2,···. More generally we have the following....

(Igusa). Prove that the higher Franz-Reidemeister torsion classes τ2i∈H 4i(IOutn; R) are non-trivial in a suitable stable range. 22. Cohomological str...

Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the right hand side is generated by (cert...

Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorphisms in (9)....

Define unstable (co)homology classes of GL(n, Z). In particular, what can be said about the image of µi ∈ H4i(OutF2i+2; Q) in H4i(GL(2i + 2, Z); Q) un...

Compute the cohomology of AutFn and OutFn with coefficients in various GL(n, Q)-modules. 360 S. Morita For example, we could ask how Looijenga’s result ...

Determine whether the natural homomorphisms ˜H ∗(AutF2g; Q)−→˜H ∗(Mg,1; Q) ˜H ∗(OutF2g; Q)−→˜H ∗(Mg,∗; Q) induced by the inclusions Mg,1→AutF2g, Mg,∗→...

The classes e1,t 3,t 5,··· are all non-trivial. Furthermore they are linearly independent and form a basis of H 2(hQ g,1)Sp....

Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial....

Determine the image as well as the cokernel of the homomorphism (15) explic- itly. Note that Hain [ 29] proved that the image of (15), after tensored ...

Describe the Galois images in hg,1⊗ Zℓ. The above result was proved by analyzing the number theoretical enhancement of the Johnson homomorphism where ...

Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as possible. Also compare these clas...

Study the central extension (20) from the point of view of group cohomology as well as geometric topology. In particular determine the Euler class of ...

1. ¯σ∗(˜t2k+1) is non-trivial in H 2(Hg,1) for any k 2. σ∗(˜t2k+1) is trivial in H 2(Hg,1) for any k. The first part of the above conjecture is the “g...

Determine the abelianization of the group Hg,1. Is it trivial? Also determine the second homology group H2(Hg,1; Z). Is the rank of it equal to 1 give...