10 problems

Gersten's conjecture. There is a recursive function $f\colon\mathbb{N}\to\mathbb{N}$, such that every finite presentation of a linear group has

Sharp Heisenberg eigenvalue and minimal profile. The explicit family satisfies

A group $G$ is residually free if, for every $g\in G\setminus\{1\}$, there is a homomorphism $\phi\colon G\to F_2$ to the rank-two free group such that $\phi(g)\neq 1$. For a finit…

Let $F_{m_1},\ldots,F_{m_n}$ be finitely generated free groups, and let $G$ be a finitely presented subgroup of their direct product. Its Dehn function measures the number of conju…

Let $K$ and $L$ be simply connected nilpotent Lie groups with one-dimensional centres, of nilpotency classes $k$ and $\ell$, respectively, and let their central product be … Here…

Let $H$ be a finitely presented group. Denote by $\operatorname{FA}_H$ its homological filling-area function and by $\delta_H$ its Dehn function, and write $f\preceq g$ for the usu…

Let $G$ be a finitely presented group. A function is superadditive if $f(m+n)\geq f(m)+f(n)$ for all nonnegative integers $m,n$. Write $f\simeq g$ when the two functions are equiva…

Let $\mathfrak{k}$ and $\mathfrak{l}$ be nilpotent Lie algebras of step $k$ and $\ell$, respectively, with $k\geqslant \ell\geqslant 2$, and with one-dimensional centers…

Let $\mathbf{G}$ be the algebraic group over the global field $K$ and let $S$ be the specified finite set of places, with ring of $S$-integers $\mathcal{O}_S$. Write … for the geom…

Thurston's conjecture. When $n\geq 4$, $SL(n;\mathbb{Z})$ has a quadratic Dehn function.