Mathematics Problem Archive

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable....

Consider two nested convex plane domains, and let $T_1,T_2$ be their billiard ball maps on the oriented lines intersecting both domains. If $T_1\circ ...

Let $f(x,y)$ be a polynomial for which $0$ is a nonsingular value, let $\gamma$ be an oval component of $f(x,y)=0$, and assume that $\gamma_\varepsilo...

Assume the origami hyperbolic-paraboloid pattern has invisible straight folds along a chosen diagonal of each elementary trapezoid. What is the shape ...

Find conceptual proofs and possible generalizations of the new projective-geometry configuration theorems described in the source. In particular, prov...

For a fixed combinatorial dissection of a square into $n$ triangles, the ordered triangle areas satisfy a polynomial relation depending only on the co...

Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any Baumslag-Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic? If $G$ embeds i...

Suppose G admits a finite K(G, 1), does not contain Z × Z, and whenever x ∈ G is an infinite order element such that xm and xn are conjugate, then |m|...

(Whyte) Let Γ be a 1-ended hyperbolic group. Can a finite index subgroup of Γ be isomorphic to a subgroup of Γ of infinite index?...

(Swarup) Prove the combination theorem for relatively hyperbolic groups....

(M. Mitra) Let G be a word-hyperbolic group and H a word-hyperbolic subgroup. Does the inclusion H → G extend to a continuous map between the boundari...

(Swarup) Suppose $G$ is a hyperbolic group which is a graph of hyperbolic groups such that all edge-to-vertex inclusions are quasi-isometric embedding...

(Swarup) Is there a proof of Johannson’s theorem that Out(π1M) is virtually generated by Dehn twists for M a Haken 3-manifold along the lines of Rips-...

(Exercise in [BGS85, p.2]) Take a closed surface S of genus ≥2. Let V = S × S and let Σ ⊂V denote the diagonal. Let ˜V be a nontrivially ramified fini...

(D. Wise) Let G act properly discontinuously and cocompactly on a CAT(0) space (or let G be automatic). Consider two elements a, b of G. Does there ex...

(Ross Geoghegan) A proper CAT(0) space $M$ is almost geodesically complete if there is $R\geq0$ such that for all $a,b\in M$ there is an infinite geod...

(Dani Wise) A triplane is a CAT(0) space obtained by gluing three Euclidean half-planes along their boundaries, and a CAT(0) space $X$ has isolated fl...

(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)....

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

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....

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

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

(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...

(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 ...

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

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

(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?...

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

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...

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...

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

Are 1-relator groups coherent?...

Let $\Gamma$ be a countable subgroup of $\operatorname{SL}_2(\mathbb{Q}_p)$ containing no parabolic elements, and let $\gamma$ be a random element of ...

For an infinite $d$-regular Ramanujan graph, does random-walk neighborhood sampling converge to the $d$-regular tree?...

For each $d\geq3$, does the independence ratio of a uniformly random $d$-regular graph converge in probability as the number of vertices tends to infi...

Let $G$ be an infinite Cayley graph of a group that is not virtually cyclic. Prove that there exists $p<1$ for which Bernoulli $p$-edge percolation on...

Does every infinite connected Cayley graph admit an invariant random perfect matching?...

For a nonamenable group $\Gamma$, does the Bernoulli shift $\{0,1\}^{\Gamma}$ factor onto $\{0,1,2\}^{\Gamma}$?...

Can every $d$-regular graphing without multiple edges be properly edge-colored by $d+1$ colors?...

Let an amenable group $\Gamma$ act ergodically and essentially faithfully on $X$. Is the groupoid cost of the action $1$? If $\Gamma$ is finitely pres...

In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynom...

Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse....