Mathematics Problem Archive

(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?...

Are 1-relator groups coherent?...

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 $\Gamma$ 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 $C_2$....

Can the sequence of metric balls in an infinite Cayley graph form a family of expanders?...

Suppose $G$ and $H$ are Cayley, or vertex-transitive, expanders on the same number of vertices and can be matched with asymptotically vanishing edge d...

If $G$ is a finite vertex-transitive expander, must every almost automorphism of $G$ be close to an automorphism?...

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...

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

For a locally convergent sequence $(G_n)$ of bounded-degree integer-labeled graphs, does the normalized rank modulo $p$ of the adjacency matrix conver...

For a graph sequence $(G_n)$ define $e((G_n))=\liminf |E(G_n)|/|V(G_n)|$, and define its combinatorial cost as the infimum of $e((H_n))$ over graph se...

Compactness implies that for every $\varepsilon>0$ there is $K>0$ such that every finite graph can be approximated within error $\varepsilon$ by a fin...

Let $(G_n)$ be a locally convergent graph sequence and let $\mu_n$ be the probability distribution of the roots of the chromatic polynomial of $G_n$. ...

Which probability measures can occur as eigenvalue distributions of finite $d$-regular graphs? Find natural restrictions....

Let $G$ be a $d$-regular Cayley graph with spectral measure $\mu$. Is $\mu$ a weak limit of spectral measures of finite $d$-regular graphs?...

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...

Do uniformly random $d$-regular graphs converge, in local-global convergence, to the weak closure of independent identically distributed processes?...

Is the i.i.d. action of the free group $F_2$ a local-global limit of finite actions of $F_2$?...

Let $\Gamma$ have property (T), and let $(G_n)$ be a sofic approximation of a Cayley graph of $\Gamma$. Can $(G_n)$ be changed by an asymptotically va...

Can every ergodic unimodular random network that is almost surely an infinite tree be obtained as the limit of an expander family?...

Let $G$ be an infinite vertex-transitive graph, let $A$ be a finite vertex set, let $b$ be a vertex, and let $\partial A$ be the set of vertices at di...

Define the first $L^2$ Betti number of a vertex-transitive graph $G$ from the expected degree of a free spanning forest. Do $G$ and its square $G^2$ h...

Can free spanning forests be used to prove basic properties of the first $L^2$ Betti number, such as multiplicativity upon passage to a finite-index s...

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?...

Does every infinite Cayley graph, or every infinite vertex-transitive graph, have a spanning tree without leaves?...

In every infinite Cayley graph, does the density of dead ends tend to zero?...

Are factors of i.i.d. on the $3$-regular tree closed in the weak topology? In particular, is the weak limit of majority functions on $n$-balls a facto...

Does every infinite Cayley graph $G$ admit a $G$-invariant proper coloring with $\chi(G)$ colors?...

Let $X$ be the space of $k$-regular Cayley graphs with the local-convergence topology and let $T\subset X$ be the closed subset of transient graphs. I...

For every $k>1$, does there exist $C(k)<1$ such that the return probability of every transient $k$-regular Cayley graph is at most $C(k)$?...

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 $(G_n)$ and $(H_n)$ converge to the same graph limit, and let $(T_n)$ be a convergent sequence with each $T_n$ a spanning tree of $G_n$. Do there ...

Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that can be properly colored by $C$ colors with arbitrarily small error. Can it be...

Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that weakly contains a finite graph $H$. Does $G$ factor onto $H$?...

Does every higher-rank semisimple real lattice have rank gradient zero?...

Does every residually finite property-(T) group have rank gradient zero?...

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 $\operatorname{Aut}(F)$?...