6 problems

Grigorchuk–Lubotzky–Pyber conjecture. The two asymptotic constants coincide and equal

Let $B_n$ be the braid group on $n$ strands, and let $a_k(B_n)$ denote the number of subgroups of index $k$ in $B_n$. For $n$ sufficiently large, write $c(k/n)$ for a constant depe…

For a finite group $G$, let $\rho_p(G)$ denote its $p$-rank and let $\rho(G)$ denote its rank. For an arbitrary group $G$, let $\mathcal{F}(G)$ be the set of its finite quotient gr…

Let $G$ be a finitely generated soluble group. Its subgroup growth is said to have a subgroup growth gap if there is no subgroup growth rate strictly between polynomial growth and…

Let $G$) be a semisimple Lie group, and let $\rho_x(G)$ denote the number of lattices in $G$ of covolume at most $x$, up to automorphism. For a lattice $\Gamma$ in $G$, let…

Subexponential subgroup growth conjecture. If $G$ has subexponential subgroup growth and