31 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…

André's conjecture. There are infinitely many non-arithmetic p-adic triangle groups.

For a Euclidean triangle group with counting function $N(\lambda)$, the leading Weyl term is followed by an error term whose order is not known. Error-term conjecture. The next ter…

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 $I$ be the critical interval for the family $\rho_t(p,q,r)$, let $\Gamma$ be the abstract $(p,q,r)$ triangle group, and for an infinite word $W\in\Gamma$ write $W_t=\rho_t(W)$.…

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 $(p,q,r)$ be a triple with $p\leq q\leq r$, and let $\rho_t(p,q,r)$ be the associated one-parameter family. Call $(p,q,r)$ type A when the endpoints of the critical interval oc…

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…

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.

Chain-link ancestry conjecture. The chain link $C_p$ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group $\Delta_{p,q,r;\infty}$, for…

Let $\Delta(2,3,7)$ be the $(2,3,7)$-triangle group acting on the hyperbolic plane $\mathbb{H}$, and let $\ell_n$ denote the lengths of the relevant closed geodesics, subject to th…

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…

Let $\Delta_{3,n,m;\infty}$ be the complex hyperbolic triangle group with parameters $3,n,m,\infty$, and let $6_1^3$ denote the three-cusped hyperbolic magic 3-manifold. Dehn filli…

Exotic-manifold conjecture. Closed, exotic $4$-manifolds should appear in Artin presentation theory with second Betti number three, and this should relate to the Torelli subgroup i…

Critical-interval discreteness and faithfulness conjecture. The $(n_1,n_2,n_3)$-triangle groups are conjectured to be discrete and faithful in $SU(2,1)$ on $\mathcal{I}$.

Type classification conjecture. Otherwise, the triangle has type B. It was conjectured that $(n_1,n_2,n_3)$ has type A for $n_1<10$ and type B for $n_1>13$.

The short-word discreteness conjecture. A $(n_1,n_2,n_3)$-complex hyperbolic triangle group is a discrete embedding in $PU(2,1)$ if neither $W_A$ nor $W_B$ is non-elliptic.

Let $\Gamma$ be a discrete triangle group of type $(m_1,m_2,\infty)$, and let $G={\rm GL}(2,\mathbb{C})$. Define … Let $Z(G)\cong \mathbb{C}^*$ be the group of scalar matrices. The…

Strong automorphism realisation conjecture. The condition that at least one of $p$, $q$ and $r$ is $\infty$ can be omitted from the countable and strong automorphism realisation as…

Let $\Gamma$ be a hyperbolic triangle group. A subgroup has infinite index if its index in $\Gamma$ is infinite, and two subgroups are conjugate when one is obtained from the other…

Let $G=\Delta(p,q,r)$ be the triangle group with presentation … Assume that $p$ is prime and $p\leq q\leq r$, and choose an integer $m$ such that the alternating group ${\cal A}_m$…

Let $G=\Delta(p,q,r)$ be the triangle group with presentation … Assume that $p$ is prime and $p\leq q\leq r$. Write $C_p$ for the cyclic group of order $p$, and let…