47 problems

Let $\rho$ be a complex hyperbolic ultra-parallel $[0,0,0]$-triangle group representation, with generators $R_1,R_2,R_3\in\mathrm{PU}(2,1)$. Goldman–Parker conjecture. Such a repre…

Schwartz's conjecture. The $(p,q,r;\alpha)$ representation of $\Delta_{p,q,r}$ is discrete and faithful if and only if $W_A$ and $W_B$ are non-elliptic. Furthermore, if $p<10$, it…

Schwartz's conjecture. The representation is discrete and faithful if and only if neither $W_A$ nor $W_B$ is elliptic.

Let $\Delta(p,q,r;\theta)\subset \operatorname{PU}(2,1)$ be the complex hyperbolic triangle group generated by complex reflections $I_1,I_2,I_3$, and let … be the corresponding abs…

A WDVV solution is a solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations, and it has the required good analytic properties when it satisfies the analytic conditions intend…

Margulis's conjecture. The group $\Gamma$ is an arithmetic lattice in $G$.

Schwartz's conjectural picture. A $(p_1,p_2,p_3)$-representation is a discrete embedding if and only if neither $w_A$ nor $w_B$ is elliptic. The corresponding parameter values form…

Let $(p,q,r)$ be a triple with $p\leq q\leq r$, let $\rho_t(p,q,r)$ be the associated family, and classify the triple as type A or type B according to the critical-interval definit…

Let $\rho_t(p,q,r)$ be the one-parameter family of representations of the $(p,q,r)$-reflection triangle group by complex reflections, with $p\leq q\leq r$ and $\rho_0$ stabilizing…

Let $\Gamma$ be a discrete cocompact subgroup of $SL_2(\mathbb C)$, and let … An elliptic curve is a complex curve of genus one. Two elliptic curves are non-isogenous when there is…

Let $G$ be a Lie group, let $H$ be a reductive subgroup, and let a tessellation of $G/H$ mean a properly discontinuous, compact-quotient action by a discrete subgroup of $G$. Suppo…

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname F_n$, $G_1$ and $G_2$ are of type $\operatorname F_{n+1}$, and $Q$ is of type $\operatorname F_{n+2}$, then…

Let $2\leqslant k\leqslant m$ be integers. For groups $G_{1},\ldots,G_{m}$, a subgroup $P\leqslant G_{1}\times\cdots\times G_{m}$ virtually surjects on $k$ factors if, for every…

Let $G/P$ be a flag manifold, where $G/P$ is neither real projective space nor the conformal sphere. A proper divisible domain is a proper domain in $G/P$ admitting a properly disc…

Schwartz's conjecture. The group $\Delta(p,q,r)$ is discrete and faithful if and only if both $W_A$ and $W_B$ are non-elliptic.

Measure-sequence characterization. The sequences of measures defining a density in this sense are precisely those satisfying

Let $G$ be a discrete group. Its amenable radical is the largest normal amenable subgroup of $G$, and $G$ is $C^*$-simple when its reduced $C^*$-algebra has no non-trivial closed t…

Let $G/H$ be a homogeneous space of reductive type. Say that $G/H$ satisfies (P-free) if it admits a proper action of a non-abelian free discrete subgroup of $G$, and satisfies (P-…

Let $G_0$ be a real linear simple Lie group of real rank at least $2$, let $K$ be a maximal compact subgroup of $G_0$, and let $\Gamma$ be a discrete subgroup of $G_0\times G_0$ ac…

Congruence subgroup property conjecture. Lattices in $G$ have this slightly weaker form of the congruence subgroup property if and only if

Let $\Delta(p,q,r)$ be the abstract $(p,q,r)$ reflection triangle group, and let $\Gamma=\Delta_{\infty,\infty,\infty}=\langle I_1,I_2,I_3\rangle$ be a complex hyperbolic ideal tri…

Finite presentability conjecture. Then $H$ is finitely presented.

Let the triangular and hexagonal superpackings be the packings discussed in the source, and let their symmetry group be the group referred to immediately before the conjecture. A p…

Discrete periodic tiling conjecture. Let $F$ be a finite non-empty subset of a finitely generated discrete abelian group $G$. Then the tiling equation $A\oplus F=G$ is not aperiodi…

Let $G$ be a simple non-compact real algebraic group and let $H<G$ be a proper simple non-compact subgroup. Consider the question of finding sufficient conditions ensuring that eve…