16 problems

Let $(A,u)$ be a finitely generated quantum group, and let $b_k$ and $p_k$ denote the growth and random-walk quantities constructed in the paper. Here $f_k\approx g_k$ means that t…

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and let $A_{n,k}$ denote its $k$-th consecutive quotient of the lower central series. Write…

Let $A(N)$ denote the area covered by a prime walk up to the explored interval of length $N$, namely the number of unique lattice points $(x,y)$ visited by the walk. Area divergenc…

Let $A$ be a Jordan PI-algebra, and let its codimension growth be measured by the sequence of ordinary codimensions. The scale 2 referred to in the paper is the scale for codimensi…

Let $L$ be a finitely generated Lie PI-algebra. Write $\gamma_L(n)$ for its growth function, and let $\ln^{(q)}n$ denote the $q$-fold iterated natural logarithm. Intermediate-growt…

Let $F_r$ be a free group and let $w\in F_r$ be a non-primitive element. Define its automorphic orbit by … The growth rate of the set of all conjugates of $w$ is $\sqrt{2r-1}$. Pud…

Let $F_r$ be a free group and let $w\in F_r$ be a non-primitive element. Let $CA(w)$ denote the set of cyclically reduced automorphic images of $w$. Cyclically reduced orbit growth…

Let $G$ be a connected Lie group. The notions of algebraic and analytic thickness and thinness are those used in the paper, and Theorem is the thick/thin dichotomy for unimodular c…

Let $G$ be a finitely generated, contracting, regular branch group, and let $(a_n)_{n=0}^{\infty}$ denote its portrait growth sequence. Portrait growth conjecture. The portrait gro…

Let $\widetilde{\mathcal{H}}_n$ denote the class of elements considered in the paper, and let $A\in\widetilde{\mathcal{H}}_n$ have infinite order. The exponential core growth conje…

Let $G\subset U_N^+$ be a compact quantum group, and let $\widehat{G}$ denote its discrete dual. For each $Q\in U_N$, let $\Gamma_Q$ be the associated discrete quantum group. Polyn…

A Koszul algebra is a graded algebra with the Koszul property. Let $A$ be a Koszul algebra, and let its Hilbert series be $H_A(t)$. Rational Hilbert series conjecture. The Hilbert…

Let $S$ be a left reversible semigroup if any two principal left ideals intersect, and call $S$ finitely generated when a finite subset generates it. Its growth is polynomial when…

Let $G$ be an infinite finitely generated group, and let $\Gamma_n(G)$ denote its product replacement graph, whose vertices are generating $n$-tuples of $G$ joined by Nielsen moves…

Let $H$ be a finite-index subgroup of a finitely generated group $G$. The finite-index subgroup conjecture asserts that if $\Gamma_n(G)$ has exponential growth, then $\Gamma_k(H)$…

Let $G$ be an infinite finitely generated group. The exponential Nielsen growth conjecture asserts that there are an integer $n$ and a generating $n$-tuple $S\in\Gamma_n(G)$ such t…