Mathematics Problem Archive

(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(\...

(Yves de Cornulier) Let G be residually torsion-free nilpotent. Is G Haagerup (= a-(T)-menable)?...

(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?...

(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 $C_2$....

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?...