Abstract Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $$\text {PGL}_2(\mathbb {F}_q)$$ PGL 2 ( F q ) or $$\text {PSL}_2(\mathbb {F}_q)$$ PSL 2 ( F q ) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.
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Borel and Volume Classes for Dense Representations of Discrete Groups
Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.
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- Award ID(s):
- 1902896
- NSF-PAR ID:
- 10466055
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2022
- Issue:
- 15
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 11891 to 11956
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab078
