20 problems

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 … A one-relator group is a group admitting a presentation with one defining relator. Gersten's conjecture. The group $G_{(1,2)}$ has the highest Dehn function among all one-rel…

An S-machine is a rewriting device whose rules form its program; its Dehn function measures the maximal area of a van Kampen diagram of boundary length at most a given value. Olsha…

Let $G'_N(\mathcal S)$ be a hub-free realization of an $S$-machine, where $\mathcal S$ is an $S$-machine and $N$ is the associated parameter. Rips and Sapir's conjecture. Some grou…

Let $BS_{k,1}=\langle a,b\mid a^b=a^k\rangle$ for $k\ge 2$, where $x^y=y^{-1}xy$. A 1-relator group is a group admitting a presentation with one defining relator, and its Dehn func…

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…

Improved embedding conjecture. For every $\varepsilon>0$, the group $G$ can be embedded into a finitely presented group $H$ whose Dehn function is bounded above by…

Let … Here $I_N$ is the special one-relation inverse monoid associated with the monoid $\Pi_N$. Conjecture on $I_N$. The Dehn function of $I_N$ is at least exponential, and the wor…

Superadditive-closure conjecture. Every Dehn function is equivalent to its superadditive closure:

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 $\mathfrak{k}$ 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…

Gromov's conjecture. If $X$ has rank at least $3$, then every irreducible lattice $\Gamma\subset\operatorname{Isom}(X)$ has polynomial Dehn function.

Let $G$ be a symmetric space of rank $k$, and let $\Gamma$ be a lattice in $G$. The $(k-2)$-th order Dehn function of $\Gamma$ measures higher-dimensional filling complexity. Gromo…

Let $\Gamma$ be an $S$-arithmetic subgroup of a reductive algebraic group defined over a global field $\mathbb K$ such that $\Gamma$ acts as an irreducible non-uniform lattice on…

Let $G$ be an $HF^{\infty}$ group, let $\mathcal{P}$ denote the polynomial bounding class, and let the higher Dehn functions and first Dehn function of $G$ be defined as in the sou…

Let $B$ be the group in presentation, with $n \gg 1$ chosen as in Theorem1.10, and let $f_{1,B}(x)$ denote its first Dehn function. For odd $n>10^{10}$, the verbal Dehn function of…