33 problems

Let $G=\langle X\rangle$ be a group generated by a finite set $X$, with growth function … Write $f\prec g$ for the growth comparison used in the source: there is a constant $C>0$ s…

Let $G$ be a finitely generated group. Milnor's polynomial growth conjecture. The group $G$ has polynomial growth if and only if it contains a nilpotent subgroup of finite index. G…

Let $G$ be an ordinary finitely generated group, and compare its standard growth with its conjugacy growth. Guba–Sapir conjecture. Exponential standard growth of $G$ implies expone…

Let $G$ be a finitely generated group. It is scale-invariant if it has a descending chain $\{G_k\}_{k\geq 0}$ of finite-index subgroups such that each $G_k$ is isomorphic to $G$ an…

Gap Conjecture. If a finitely generated group $\Gamma$ has growth type strictly less than $e^{\sqrt{n}}$, then $\Gamma$ is virtually nilpotent.

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ satisfy $0<\beta<1$. Gap Conjecture with parameter $\beta$. If … then $G$ has polynomial g…

Let $f$ be a quadratic polynomial whose critical orbit is preperiodic, and let its iterated monodromy group be the group obtained from the monodromy actions on the successive itera…

Let $\ThePolynomial=\TheComplex^2+\TheConstTerm$ be a quadratic polynomial, and let $\operatorname{IMG}(\ThePolynomial)$ denote its iterated monodromy group acting on the infinite…

Exponential conjugacy-growth conjecture. The conjugacy growth of $G$ is exponential.

A group is Choquet–Deny if every bounded harmonic function is constant for every probability measure on it. A group has exponential growth if the size of balls in a finitely genera…

Let $G$ be a group generated by a finite set $X$, and let $\gamma_X(n)$ denote its growth function. For functions $f,g:\mathbf{N}\to[0,\infty)$, write $f\preceq g$ if there is a co…

Let $d e 0$, let $G$ be a finitely generated infinite residually finite group with uniformly $(1/d)$-almost flat quotients, and let $H$ be a finite-index subgroup admitting a quoti…

Let $G$ be a finitely generated residually finite group. Having uniformly almost flat quotients means satisfying the quotient condition defined in the paper, and having uniformly a…

Weak gap conjecture. If $G$ has growth $\preceq r\mapsto\exp(r^\alpha)$ for some $\alpha<\tfrac12$, then there exists a Banach metric $d$ on $G$ such that $\partial(G,d)$ contains…

Subexponential-growth finite-orbit conjecture. The horofunction boundary of $Gamma$ contains a finite orbit for the underlying group action.

Pittet–Saloff-Coste conjecture. Conversely, this decay should characterise groups of finite rank among torsion-free solvable groups. The conjecture asks whether the stated return-p…

Quadratic Schreier growth gap conjecture. $G$ has a Schreier growth gap $n^2$.

Let $G$ be a finitely generated group that is not virtually nilpotent, and let its growth function count the elements representable by words of length at most $n$. Subexponential g…

Let $G$ be a group with finite generating set $S$, and let $B_n(S)$ denote the set of elements of $G$ representable by words of length at most $n$ over $S$. A group has polynomial…

Grigorchuk's modified Milnor problem. A finitely presented group with vanishing algebraic entropy has at most polynomial growth.

Let $\Gamma$ be a finitely generated group. Call $\Gamma$ strongly-connectivity-forcing if, for every finite generating set $S$ with $\operatorname{Id}\notin S$ and $S^{-1}=S$, and…

Let $\mathbb{G}$ be a finitely generated group, and let $\mu$ be a symmetric probability measure on $\mathbb{G}$ with finite support that generates $\mathbb{G}$. Let…

Let $G$ be a finitely generated group such that $h_{alg}(id_G)=\infty$, equivalently, $G$ has exponential growth. Infinite algebraic entropy conjecture. Then $h_{alg}(\phi)=\infty$…

Let $K$ be a field, and let $A$ be a finite subset of $\operatorname{GL}_n(K)$ satisfying $A=A^{-1}$ and $e\in A$. For every $R\geq 1$, consider the alternatives involving the trip…

Let $G$ be a finitely generated amenable group, and let $P(n)$, $\mathcal F(n)$, $\Lambda(n)$ and $\mathcal N(\lambda)$ denote respectively the heat-kernel, Følner, isoperimetric-p…