17 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 $d e 0$, 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…

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

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

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

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…

Let $G$ be a finitely generated amenable group, with the heat-kernel function $P(n)$, Følner function $\mathcal F(n)$, isoperimetric profile $\Lambda(n)$, and spectral density…

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ be the parameter used in the stronger form $G^*(\beta)$ of the parameterized Gap Conjectur…

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ satisfy $0<\beta<1$. Weak Gap Conjecture. There exists such a $\beta$ for which … implies…

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 $G$ be a finitely generated group with growth function $\gamma_G(n)$. Write $f\prec g$ when $f$ is strictly bounded above by $g$ in the relevant growth comparison. Gap Conjectu…

Let $G$, $\mu$, $P(n)$, $F(n)$ and $\mathcal N(\lambda)$ be as in the corresponding gap conjectures. Pansu–Grigorchuk gap conjectures with parameter $\beta$. There exists a number…

Let $G$ be an amenable finitely generated group, let $\mu$ be a symmetric finitely supported probability measure generating $G$, let $M$ be the associated Markov operator, and let…

Let $G$ be a finitely presented group. A free subsemigroup on two generators is a subsemigroup generated by two elements with no nontrivial relations. Grigorchuk's stronger conject…