40 problems

Let $H$ be a coherent hyperbolic group, and let $G$ be an extension of $H$ by $\mathbb{Z}$. Wise's coherence conjecture. The group $G$ is coherent. The source attributes this conje…

Let $A$ be a connected regular algebra of exponential growth, where regularity means finite global dimension together with the stated Gorenstein property. Bondal's coherence conjec…

Surface-by-cyclic group-algebra coherence conjecture. The group algebra $\mathbb{Q}[G]$ is coherent. The source identifies the virtual surface-group case as difficult and open, mot…

Let $\rho_{\mathrm{mc}}=\sum_{i,j}\rho_{ij}\lvert ii\rangle\!\langle jj\rvert$ be a maximally correlated state, and let…

Higher-dimensional coherence conjecture. Any small lax $n$-category is laxly $n$-equivalent to a small Gray $n$-category.

Coherence conjecture for semidirect sums with a one-dimensional Lie algebra. If

Lie-algebra Kropholler–Walsh–Wise conjecture. If

Let $G$ be a virtually locally indicable group. A group is coherent if all its finitely generated subgroups are finitely presented, and a group is of type…

Let $G$ be a group of geometric dimension at most $2$. A $2$-complex has nonpositive immersions if every compact, connected $2$-complex immersing cellularly into it has nonpositive…

Let $G$ be a RFRS group of type $\mathrm{FP}$. A group is coherent if every finitely generated subgroup is finitely presented. The algebra $\mathbb{Q}[G]$ is the rational group alg…

Coherence conjecture. All such maps defined as composites of associators, unitors, and interchange morphisms are equal.

Non-coherence conjecture. For all $0 \leq m \leq n-1$, $G$ is not $m$-coherent.

Let a theory be quotient by a subtheory $\W$, and call the subtheory rigid when all diagrams in it commute. Coherence conjecture. When the subtheory is rigid, there is always coher…

Higher incoherence conjecture for poly-free groups. All poly-free groups exhibit the same higher incoherence properties as direct products of free groups.

Let $A$ be the right-angled Artin group for which Bridson constructed a finitely presented subgroup of $A\times A$ with undecidable conjugacy and membership problems. A right-angle…

A compact $2$-complex $X$ has nonpositive immersions if every compact, connected, collapsed combinatorial immersion $Y\rightarrow X$ has either trivial fundamental group or satisfi…

Let $X$ be an aspherical two-complex and let $G=\pi_1(X)$ be its fundamental group. Say that $G$ has trivial second $L^2$-Betti number when $b_2^{(2)}(G)=0$. Betti coherence conjec…

A group is coherent if every finitely generated subgroup is finitely presented. A one-relator group is a group admitting a presentation with one defining relator. Baumslag's conjec…

Non-positive irreducible curvature implies coherence. If $\rho_+(X)\leq 0$ then $\pi_1(X)$ is coherent.

Bottleneck conjecture. Moving beyond the $k^2$ bottleneck is likely to require tools beyond the notion of $alpha$-coherence.

Let $F$ be a frame for $\mathbb{F}^n$, and let $G$ be one of its dual frames. Write $Gr(F,G)$ for their cross Gramian and let $\mu(Gr(F,G))$ denote its maximum off-diagonal magnitu…

Conjecture that Bridson's group is not coherent. The right-angled Artin group $A$ in Bridson's example cannot be coherent.

Let $\mathcal{C}$ be a braided semigroupal polyadic category with an $n$-ary tensor product, an $n$-ary associator $\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes}$…

$n$-ary coherence conjecture. If the $n$-ary associator $\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes}$ satisfies such $n$-ary coherence conditions that this isom…

Let $G=F_m\rtimes F_n$ be a semidirect product of free groups, where $m,n\geq 2$. A group is incoherent if it has a finitely generated subgroup that is not finitely presented. Wise…