Mathematics Problem Archive

(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)?...

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...

(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...

(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....

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

(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...

(Grigorchuk) Aut(Fn) acts on the space ∂3Fn of triples of distinct ends of Fn. Denote by Yn the compact space (Cantor set) the quotient space of ∂3Fn ...

Do there exist f.g. free subgroups of MCG(S) consisting of identity and pseudo-Anosov mapping classes which are not Schottky?...

Do there exist non-free pseudo-Anosov subgroups?...

For $\operatorname{Out}(F_n)$, do there exist finitely generated free subgroups consisting of the identity and irreducible automorphisms which are not...

Does every automorphism h : Fn →Fn leave invariant a finite index subgroup K such that hab : K/K′ →K/K′ has an eigenvalue which is a root of unity?...

Does every closed 3-manifold M which fibers over S1 with fiber of genus ≥2 have a finite cover ˜ M with b1( ˜ M) > 1?...

Is MCG(Sg) →QI(MCG(Sg)) an isomorphism for g ≥3?...

Suppose that φ : MCG(Sg) →MCG(Sg) is a quasi-isometry. Does φ map maximal flats to maximal flats (“maximal flats” come from maximal rank abelian subgr...

For $\operatorname{Out}(F_n)$, is the natural map $\operatorname{Out}(F_n)\to QI(\operatorname{Out}(F_n))$ an isomorphism? Does every quasi-isometry o...

If Γ is an irreducible uniform lattice in a higher rank connected semisimple Lie group, does every homomorphism Γ →Out(Fn) necessarily have finite ima...

Is MCG(Sg,b,n) linear?...

Let $SB_n$ be the singular braid monoid generated by $\sigma_i^{\pm1}$ and singular generators $\tau_i$. Define $\Phi:SB_n\to\mathbb{Z}B_n$ by $\Phi(\...

(Kevin Whyte) Let K be a finite complex with π = π(K) amenable. Is there a uniform bound to the betti numbers of finite covers of K?...

(Kevin Whyte) Is every solvable PD(n) group polycyclic?...

(S. Ivanov) Is there a f.p. slender group which is not polycyclic-by-finite?...

(Seymour Bachmut) Is SL2(K) finitely generated for K = Z[X, X−1] or K = F[X, X−1, Y, Y −1] for a field F?...