Mathematics Problem Archive

(Henry Glover) Does for every finite graph G the following 1−2−∞- conjecture hold: G is planar, a double cover of G is planar, or no finite cover is p...

(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?...

Are 1-relator groups coherent?...

Let $L$ be a flag triangulation of $S^{2k-1}$, let $f_i$ be the number of $i$-simplices of $L$, and define $$\chi=1-\sum_{i=0}^{2k-1}(-1)^i\frac{f_i}{...

Do there exist groups G with balanced presentation (same number of generators and relations), with H1(G) = 0 and with unsolvable word problem?...

Is there a sequence of (perfect, of course) groups with balanced presentations among which one cannot recognize trivial groups?...

Let $G$ be a one-relator group whose relator $W$ is a cyclically reduced word in the generators, and let $P$ be the submonoid of $G$ generated by all ...

Can the odometer acting on the rooted binary tree be embedded in a nonabelian free pro-$2$ group?...

Let a finitely presented pro-$p$ group have $r+2$ generators and $r$ relators. Must it have an open subgroup that maps continuously onto a nonabelian ...

Let $G$ be a closed transitive subgroup of the automorphism group of a rooted tree. Must every level stabilizer of $G$ contain an element acting witho...

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 ...

Determine the asymptotic length of a shortest nontrivial group law for the $n$-fold iterated wreath product of $C_2$....

Let $\Gamma$ be finitely generated and let $\{H_n:n\geq1\}$ be a property-$\tau$ family of finite-index normal subgroups. Does the chain $\Gamma_n=\bi...

Let $\Gamma$ be finitely presented with a chain of finite-index normal subgroups having trivial intersection and property $\tau$. Must $\Gamma$ contai...

Does every countable group have fixed price; that is, do all its free probability-measure-preserving actions have the same cost?...

Let $\Gamma$ act on $X$ by probability-measure-preserving maps and let $H\leq\Gamma$ have finite index. Is $\operatorname{cost}(H,X)-1=(\operatorname{...

For a free action of $\Gamma$ on $X$, is $\operatorname{cost}(\Gamma,X)=\operatorname{cost}(\Gamma,X\times X)$ for the diagonal action on $X\times X$?...

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...

Given a graphing of an equivalence relation, is there a subgraphing whose cost is arbitrarily close to the cost of the relation?...

Can a nonabelian free group $F$ have a nontrivial pseudocharacter invariant under $\operatorname{Aut}(F)$?...

Are two independent random subsets of $\mathbb{Z}$ almost surely quasi-isometric as metric spaces? What is the answer for other Cayley graphs, such as...

Let $P$ be a finite $p$-group of order $n$ and let $w$ be any word. Is the probability that $w$ is satisfied in $P$ at least $1/n$?...

Let $M$ be the set of measurable real functions, and define $H_f(x,y)=(x,y+f(x))$ and $V_f(x,y)=(x+f(y),y)$. What group is generated by $\{H_f:f\in M\...

Determine the asymptotic length of the shortest nontrivial word that is a law in every group of order $2^n$....

In Gromov's density model with generators $a,a^{-1},b,b^{-1}$, choose $3^{nd}$ relators independently and uniformly from the reduced words of length $...

Let $\beta$ be a nondegenerate bilinear form on a finite-dimensional real vector space $V$, and let $G=\operatorname{Isom}(\beta)\subset GL(V)$. Which...

Which Kleinian groups admit closed quasi-Fuchsian surface subgroups?...

The CAT(0) cubical dimension of a group $G$ is the least dimension of a CAT(0) cubical space on which $G$ acts geometrically. Is there a sequence of c...

Do closed hyperbolic $3$-manifold groups have finite-index subgroups that embed in a word-hyperbolic reflection group?...

Does every freely indecomposable cubulated hyperbolic group contain the fundamental group of a closed hyperbolic surface?...

If $M$ is a closed $3$-manifold, when does $\pi_1M$ act on the affine building for $SL(4,\mathbb{R})$ so that the quotient retracts to $M$?...

Let $\Gamma$ be a discrete group acting properly discontinuously by affine transformations on $\mathbb{R}^5$. Is $\Gamma$ virtually an extension of a ...

Classify all properly discontinuous affine actions of a given closed surface group on $\mathbb{R}^6$....

For a given right-angled Coxeter group, what is the least $n$ for which it admits a proper affine action on $\mathbb{R}^n$?...

Let $\Gamma$ be a Kleinian group of finite covolume, and let $\mathcal{C}(\Gamma)$ be the set of isomorphism classes of its finite quotient groups. Do...

For $\Gamma=F_r$ with $r\geq2$, does $\mathcal{C}(F_r)$ determine $F_r$ up to isomorphism?...

Let $\Gamma$ be a Kleinian group of finite covolume. Does its set $\mathcal{C}(\Gamma)$ of finite quotient isomorphism classes determine $\Gamma$ amon...

If $G$ is a Gromov-hyperbolic group whose boundary is homeomorphic to the limit set of a convex-cocompact Kleinian group, is $G$ virtually a convex-co...

Which subsets of $S^2$ can occur as limit sets of convex-cocompact Kleinian groups?...

For which Kleinian groups does the limit set contain a Sierpiński carpet? For which Kleinian groups does the limit set contain a continuum?...

Characterize the limit sets of graph Kleinian groups and iterated graph-Kleinian groups. A graph Kleinian group is a convex-cocompact Kleinian group f...

Can there be a degenerate Kleinian group that is not the fiber of a fibration and has algebraic trace field?...

Is there a singly degenerate Kleinian group for which all matrix entries of all group elements lie in one fixed number field?...