15 problems

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 $(p,q,r;\alpha)$ 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…

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…

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 $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$. For a positive integer $m$, let $j_m(G)$ be the dimension of the subvariety of eleme…

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…

For each integer $r\geq 1$, let $\mathcal{T}(r)$ denote the set of $r$-arithmetic triples. Nonemptiness conjecture. The set $\mathcal{T}(r)$ is nonempty for all $r\geq 1$. The prec…