21 problems

Analytic continuation conjecture. The function $\zeta_{\mathrm{ts}}(s;G,H)$ admits an analytic continuation to the complex plane, with poles reflecting the arithmetic structure of…

Let $G$ be a finite insoluble group satisfying the condition $(\mathcal{N},21)$, meaning that $G$ has 21 elements such that every pair generates a non-nilpotent subgroup. A finite…

Non-splitting conjecture. The quotient homomorphism $\operatorname{Aut}(\pa)\to \operatorname{Aut}(\pa/Z)$ has no splitting.

Clifford extension splitting conjecture. The Clifford extension splits if and only if $4\nmid |A|$.

For finite groups $H_1,H_2$, write $H_1\implies H_2$ when every finite group having a quotient isomorphic to $H_1$ also has a subgroup isomorphic to $H_2$. The symmetric-group quot…

Modular distribution of invariants. The invariants of topo-symmetric extensions are distributed uniformly modulo $\gcd(|G|,|H|)$ in the sense above. Numerical experiments for small…

Compatible normal series conjecture. Two groups are compatible if and only if they have compatible normal series.

Let $G$) be a non-elementary hyperbolic group, let $C$ be a finite subgroup satisfying the hypotheses of Theorem 1.10, and let … Thus, $G'$ is the HNN extension of $G$ over the ide…

Baumslag–Wise conjecture. Every one-relator group with torsion is virtually free-by-cyclic, and every hyperbolic one-relator group is virtually free-by-cyclic.

Let $\mathbb{TD}_n$ denote the $n$-dimensional T-duality object, and consider its automorphism 2-group $\mathrm{AUT}(\mathbb{TD}_n)$, whose homotopy groups fit into the extension d…

Let $F$ be a free group, let $H\leq K\leq F$, and consider the morphism between their corresponding core graphs for every base. The extension is called algebraic when $H$ is not co…

Let $F$ be a free group, let $H\leq K\leq F$, and for each base of $F$ consider the morphism between the corresponding core graphs. Call the extension algebraic when $H$ is not con…

Throughout, let $({\Sigma},{\theta})$ be a topological mixing topological Markov chain, let $(X,T)$ be a topological transitive $G$-extension, and let…

Let $\calf$ be a saturated fusion system over a $p$-group $S$ such that $O_p(\calf)=1$. Set $\cale=F^*(\calf)$, a fusion subsystem over $T\nsg S$, and let $\call$ be a centric link…

Let $q$ be an odd prime power, let $G=\operatorname{PSL}_2(q)$, and let $\rho$ be the natural permutation representation of $G$ of degree $q+1$ on the projective line over…

Let $G$ be a split extension given by … where $H$ and $K$ are finitely generated, and let $\nu:K\to G$ be a left transversal with $\nu(1_K)=1_G$. For subsets of the character spher…

Let $1\to H\to G\to K\to 1$ be a finite extension, with $H$ and $K$ finitely generated, and let $\nu$ be a left transversal for $K$ in $G$. Write $\partial_{\infty}Fix\hat{\nu}$ fo…

Let $\left(\Sigma\times G,\sigma\rtimes\Psi\right)$ be an irreducible group-extended Markov system with finite alphabet $I$, and let $\varphi:\Sigma\rightarrow\mathbb{R}$ be a Höld…

Let $(\widetilde{G^*},A^*,G^*)$ be in Situation $(0)$, and let $B$ be the parameter set used to define the connected components. Suppose … Let $H$ be a $B$-invariant subgroup of…

Connected-component inclusion conjecture. Then

Let $C_n$ be a cyclic group of order $n$ and let $L$ be a finite-rank integral lattice. For a split group extension … let $\{E_*,d_*\}$ denote its Lyndon–Hochschild–Serre spectral…