Mathematics Problem Archive

As in 17.57, let $r(m)=\{r+km\mid k\in\mathbb Z\}$ for integers $0\leqslant r<m$; for $r_1(m_1)\cap r_2(m_2)=\emptyset$ let the class transposition $\...

Suppose that $U$ is a nonprincipal ultrafilter on $\omega$, and $B$ is a group such that every element $b\in B$ belongs to a subgroup of $B$ that is a...

Suppose $\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\to H$, there exists an endomorphism $\beta_f$...

Suppose B is a subgroup of the symmetric group $S_\Omega$ on an infinite set $\Omega$. Will the amalgamated free product $S_\Omega *_B S_\Omega$ of tw...

Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\times A_2\times A_3\to G$ is bijective. Mus...

For a finite group G, the power graph P(G) is the graph with vertex set G and edges \{x, y\} for all $x\ne y\in G$ such that either $x\in\langle y\ran...

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....

We say that a product $XY=\{xy\mid x\in X,\ y\in Y\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\in XY$ there are unique $x\in X$, ...

Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\cong G/A$ such that $A$ contains an element $a$ ...

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...

(Thurston) Is every closed hyperbolic 3-manifold finitely covered by one that fibers over the circle?...

(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 ...

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...