Mathematics Problem Archive

(Hanna Neumann Conjecture) If A and B are nontrivial subgroups of a free group, then rk(A ∩B) −1 ≤(rk(A) −1)(rk(B) −1)....

(Swarup) If G is a Fuchsian group, define area(G) to be the area of the convex core of H2/A. Since area(A) = 2π(rk(A)−1) for Fuchsian groups which are...

(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?...

Conjecture. Limit groups are CAT(0)....

Characterize groups of the form Fm ∗Z Fn which are limit groups....

Characterize 1-relator groups which are limit groups....

Conjecture: for $n>1$, the set $$\{(x_1,\ldots,x_n)\in F_n^n:x_1,\ldots,x_n\text{ is a basis of }F_n\}$$ is not definable....

Conjecture. The only definable subgroups of Fn are cyclic and the entire group....

For $x\in[F_n,F_n]$, define its genus as the least $g$ such that $x$ is a product of $g$ commutators. Let $f(g)$ be a uniform bound, independent of $x...

(de la Harpe) Is PSL2(R) a maximal closed subgroup of Homeo+(S1)?...

Is there a proper closed subgroup of Homeo+(S1) that acts transitively on (unordered) 4-tuples? Or k-tuples (k ≠ 3)? Relation to earthquakes?...

(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said abou...

(Jim Anderson) If G is a group of isometries of Hn, denote by Ax(G) the set of axes of the elements of G. If G1 and G2 are finitely generated and disc...

(Ed Taylor) Does there exist a constant c > 0 such that the limit set of every non-classical Schottky group has Hausdorff dimension ≥c....

(Misha Kapovich) Suppose $G$ is a finitely generated Kleinian group in $\operatorname{Isom}(\mathbb{H}^n)$. Is $$\delta(G)\leq\operatorname{vcd}(G),$$...

(Misha Kapovich) Is there ϵ > 0 so that δ(G) < ϵ implies that G is virtually free?...

(Misha Kapovich) Is there a finitely-generated discrete subgroup of SO(n, 1) whose action on the limit set is not ergodic? Is not recurrent?...

(Ian Leary) Suppose G is virtually of type FP over the field Fp of p elements, and let g be an element of order p. Is the centralizer of g in G also v...

(Ian Leary) Is there a group of finite vcd that does not act with finite stabilizers on an acyclic complex of dimension equal to its vcd?...

(Peter Kropholler) If G is FP over the rationals, is there a bound on the orders of finite subgroups of G?...

(Swarup) Starting with a finitely presented group $G$, take maximal graph-of-groups decompositions alternately over finite and over two-ended edge gro...

(Sageev) Is there a f.p. 1-ended group G with G ∼= G∗Z?...

(Papasoglou) Is there a f.p. torsion-free group G that does not split over a virtually abelian subgroup, but has infinitely many splittings over F2?...

(Shalen) If a finitely presented group G acts nontrivially (i.e. without global fixed points) on an R-tree, does it act nontrivially on a simplicial t...

(Mohan Ramachandran) For a finitely presented group $G$, consider: (A) some finite-index subgroup of $G$ admits a nontrivial action on a simplicial tr...

(Noel Brady) Are there groups of type Fn but not Fn+1 (n ≥3) which do not contain Z × Z? All known examples contain Zn−1....

(Olympia Talelli) Is there a torsion-free group G of infinite cohomological dimension such that there is n0 with the property that if H is a subgroup ...

Can Zp∞be embedded in an FP∞-group? Or in an FP3-group? Can an Fn-group be embedded in an Fn+1-group (n ≥2)?...

Compute the asymptotic dimension of CAT(0) groups, Out(Fn), mapping class groups, nonuniform lattices, Thompson’s group. Is there a group of finite ty...

Is there a finitely presented group $G=F/N$, with $F$ a finite-rank free group, such that $$d_G\!\left(N/[N,N]\right)<d_F(N)<\infty?$$ Here $d_F(N)$ i...

Suppose $H=F/N$ is finitely presented and contains $C^n$ for every $n$. Write $G_n=H*_{C^n}H=F*F/N_n$. If $C$ is nontrivial and finite, is $$\lim_{n\t...

(Andrews-Curtis) If K and L are simple homotopy equivalent finite 2-complexes, can one transform K to L by a sequence of elementary collapses and expa...

(Wise) Is there a finite aspherical 2-complex X with π1(X) coherent and with χ(X) ≥2?...

(Igor Belegredek) Let X be a non-positively curved symmetric space. Find conditions on a group Γ so that the space of conjugacy classes of faithful di...

(Belegredek) Is there a 3-complex X (not necessarily aspherical) which is not homotopy equivalent to a 2-complex but H³(X; {G}) = 0 for all local coef...

Study the quasi-isometry group QI(Rn). How big is it?...

(Kleiner) What are the quasi-isometries of the 3-dimensional group Sol?...

(Kleiner) What are the q.i’s of the Gromov-Thurston examples of negatively pinched manifolds?...

(Feighn) Let $\phi:F_n\to F_n$ be an automorphism and let $M_\phi$ be its mapping torus. Classify these groups up to quasi-isometry....

(Bridson) Classify the mapping tori of automorphisms $\mathbb{Z}^n\to\mathbb{Z}^n$ up to quasi-isometry....

(Bridson) Is G×Z quasi-isometric to G for G =Thompson’s group? Any f.g. group?...

(Levitt) Can every measured geodesic lamination with 2-sided leaves on a non-orientable compact hyperbolic surface be approximated by a simplicial mea...

(Lubotzky) Does Out(Fn) have the congruence subgroup property?...

(Kapovich) (a) For closed orientable surfaces $\Sigma_g$ and $\Sigma_h$, is there a faithful representation $\pi_1(\Sigma_g)\to\operatorname{MCG}(\Sig...

(Bowditch) Is the Weil-Petersson metric on Teichmüller space hyperbolic? Is it quasi-isometric to the curve complex?...

(Brock) What is the rank of the Weil-Petersson metric? What is the rank of the mapping class group?...

Assuming that the fixed subgroup Fix(α) is cyclic, find a bound on the length of a generator of Fix(α) in terms of the complexity of α....

Which α preserve an order (invariant under right translations) on Fn? If α has periodic elements it cannot preserve an order. Are there other obstruct...

Does Out(Fn) (n > 2) have a right orderable subgroup of finite index?...

Can the mapping class group (or a finite index subgroup of it) of a closed surface be embedded into Out(Fn)? Into the mapping class group of a punctur...