Schwartz's discreteness conjecture for complex hyperbolic triangle groups

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Schwartz's discreteness conjecture for complex hyperbolic triangle groups

   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

   $$\Lambda(p,q,r)=\left\langle a,b,c\;\middle\vert\;a^2=b^2=c^2=e,\ (ab)^p=(bc)^q=(ca)^r=e\right\rangle$$

   be the corresponding abstract triangle group, with $I_3I_2I_1I_2$ and $I_1I_2I_3$ the specified elements of $\Delta(p,q,r;\theta)$. Schwartz's conjecture. The group $\Delta(p,q,r;\theta)$ is a discrete and injective representation of $\Lambda(p,q,r)$ in $\operatorname{PU}(2,1)$ if and only if $I_3I_2I_1I_2$ and $I_1I_2I_3$ are both not elliptic. This conjecture gives a criterion for discreteness and faithfulness of complex hyperbolic triangle-group representations; the supplied source does not state whether it has been resolved.

   source: Raphaël Alexandre, “Redundancy of triangle groups in spherical CR representations”, arXiv:2006.09089 (2021).

2. Schwartz's discreteness conjecture for complex hyperbolic triangle groups

   Let $p,q,r$ satisfy $3\leq p\leq q\leq r$. Consider a representation of the triangle group $\Gamma_{p,q,r}$ into $\operatorname{PU}(2,1)$ obtained by sending the standard generators to order-two complex reflections $I_1,I_2,I_3$. Define

   $$W_A=I_3I_2I_1I_2,\qquad W_B=I_1I_2I_3.$$

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

   This conjecture predicts that discreteness and faithfulness of these complex hyperbolic triangle-group representations are determined by the isometry types of the two words $W_A$ and $W_B$. The source attributes the conjecture to Schwartz; its resolution status is not specified in the supplied text.

   source: Arielle Marc-Zwecker, “Triangle groups in the complex hyperbolic plane and special fibers of the momentum map in PU(2,1)”, arXiv:2304.10329 (2023).