33 problems

Geoghegan's conjecture. In 1979, Geoghegan conjectured that $F$ has all four properties: it is of type $\mathrm{F}_\infty$, it has no non-abelian free subgroups, it is not amenable…

Let a group $\Gamma$ act faithfully on a set $S$. The action is of type (A) if $\Gamma$ is finitely presented, every stabilizer $\operatorname{Stab}_{\Gamma}(s)$ is finitely genera…

Let $F$ denote R. Thompson's group, a finitely presented group known not to contain non-abelian free subgroups. Geoghegan's conjecture. The group $F$ is not amenable. A considerabl…

Let $V$ be Thompson's group, and let $\left\|g\right\|$ denote word length with respect to the generating set used in the paper. A reduced diagram with $n$ leaves defines an elemen…

A finitely generated group is co-context-free if its word problem's complement is context-free. The co-context-free embedding conjecture. A finitely generated group is co-context-f…

Hopficity conjecture. $V(G)$ is Hopfian if and only if $G$ is Hopfian.

Let $BF_n(\{1\})$ and $BF_n(\mathcal{P}_n)$ be the groups associated respectively with the trivial label group and with the pure braid group $\mathcal{P}_n$ in the construction of…

Let the dendrite-Thompson group be the finitely generated Thompson-like group of homeomorphisms of the circle preserving the lamination induced from the Julia set for the complex m…

Let $F$ denote Thompson's group $F$. Nonamenability conjecture. The group $F$ is nonamenable. This is a longstanding conjecture; the source notes that its results do not refute it,…

Let $F$ be Thompson's group acting on the unit interval. Non-amenability conjecture for Thompson's group $F$. The group $F$ is not amenable. This is a longstanding problem in geome…

Let $d\geq 2$ and $r\geq 1$, let $B_d$ be the braid group with $d$ strings, and let $H\leq B_d$ be a finitely presented subgroup. The group $bV_{d,r}(H)$ is the labeled braided Hig…

Belk–Bleak–Matucci obstruction conjecture. The group $\mathbb{Z}^d*\mathbb{Z}$ is the correct obstruction for embeddability of right-angled Artin groups in $(d-1)V$.

Non-embedding conjecture. Some of the groups $G_{,_1}$ do not embed inside Thompson group $V$.

Conjecture. $BV_{n,r}(H)$ is finitely presented whenever $H$ is finitely presented.

Let $V$ be the potential defining the vacuum state $|\b4Omega\rangle$, and let $\mathit{PSL}(2,\mathbb{Z})$ act through the representation described above. A planar perfect tangle…

Let $A$ be a right-angled Artin group, and let $n$ be the number of complementary edges in its defining graph. Belk–Bleak–Matucci's conjecture. The group $A$ embeds into $(n-1)V$ i…

Universality conjecture for coCF groups. Every coCF group embeds into Thompson's group $V$.

Let $V$ denote Thompson's group. A group is acyclic if its reduced homology with integral coefficients vanishes in every degree. Brown's acyclicity conjecture. The group $V$ is acy…

Normalish subgroup conjecture. Every normalish subgroup of $T$ contains an embedded copy of R. Thompson's group $F$ or an embedded non-abelian free subgroup.

Non-amenability criterion. If $F$ is non-amenable, then $T$ does not admit a normalish amenable subgroup, and consequently the reduced $C^*$-algebra of $T$ is simple.

Brin group embedding conjecture. Every finitely generated non-soluble subgroup $G<PL_+(I)$ contains an embedded copy of Brin's group $B$.

Let a group be co-$\mathcal{CF}$ if it has the corresponding co-$\mathcal{CF}$ property, and let $V$ denote Thompson's group. Co-$\mathcal{CF}$ embedding conjecture. Every co-…

Let $F$ be Thompson's group of orientation-preserving piecewise-linear homeomorphisms of the unit interval with dyadic-rational breakpoints and slopes powers of $2$. Geoghegan's co…

Universality conjecture for $V$. The classes $\mathcal{V}$ and $co\mathcal{CF}$ are equal:

Lehnert–Bleak–Matucci–Neuhöffer conjecture. A group has a context-free co-word problem if and only if it is a finitely generated subgroup of $V$.