8 problems

Let $F_m$ be a free group of rank $m$, let $H<F_m$ be a subgroup, and let $\Gamma_H(z)$ denote its cogrowth series for $z\in\mathbb{C}$. The rational cogrowth series conjecture. Th…

Let … be a Baumslag–Solitar group, with cogrowth series formed using the generating set $\{a^{\pm1},t^{\pm1}\}$. Baumslag–Solitar non-D-finiteness conjecture. If $M\neq N$, then th…

Let $\Gamma=\langle S\mid R\rangle$ be a group with finite generating set $S$, and let … be its cogrowth series, where $\psi_n$ counts words of length $n$ in the alphabet…

Cogrowth formula for non-negative functions. Fix $2\le c\le d$ and $3\le d$, and let $0\ne f\colon V(\mathbb{T}_{c,d})\to\mathbb{R}_{\ge 0}$ be a non-negative function on the verti…

Let $G$ be a group with finite generating set $X$. Let $C_{\mathrm{Red}}(z)$ denote the generating series counting freely reduced words over $X$ that represent the identity in $G$.…

Let $L=\mathbb{Z}_{2}\wr\mathbb{Z}$ be the lamplighter group, and suppose its cogrowth coefficients have asymptotic form … The numerical ratio analysis gives $1/2<\sigma<3/4$. Lamp…

Let $S$ be a finitely generated monoid. Its global cogrowth is maximal when the global cogrowth rate equals the cardinality of a chosen finite generating set. Left amenable means t…

Lower-bound conjecture. The quantity on the left-hand side is bounded away from zero by a positive constant depending only on $\varrho$.