Analysis of a Picard modular group

Francsics, Gábor; Lax, Peter D.

doi:10.1073/pnas.0603075103

Cited by 8 publications

(9 citation statements)

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“…We remark that Francsics and Lax, Theorem 4 of [12], prove that it suffices to consider eight isometric spheres. In nonhomogeneous coordinates (z 1 , z 2 ) these are given by…”

Section: 2mentioning

confidence: 99%

“…It follows that there are infinitely many embedded eigenvalues in the continuous spectrum, these eigenvalues are also called Maas cusp forms. This is described by Francsics and Lax in [12].…”

Section: Antiholomorphic Symmetry Of Dmentioning

confidence: 99%

“…Similarly, in [10][11][12] Francsics and Lax described a fundamental domain for the group PU(2, 1; O 1 ) = PU(2, 1; Z[i]). Their work did not include analysis of the combinatorics of the fundamental domain.…”

Section: Introductionmentioning

confidence: 99%

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The geometry of the Gauss–Picard modular group

Falbel¹,

Francsics

Parker

2010

Math. Ann.

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“…We remark that Francsics and Lax, Theorem 4 of [12], prove that it suffices to consider eight isometric spheres. In nonhomogeneous coordinates (z 1 , z 2 ) these are given by…”

Section: 2mentioning

confidence: 99%

Section: Antiholomorphic Symmetry Of Dmentioning

confidence: 99%

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The geometry of the Gauss–Picard modular group

Falbel¹,

Francsics

Parker

2010

Math. Ann.

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“…Low-quality reads were filtered using Trimmomatic-0.36 [ 53 ], and STAR [ 54 ] was used to map filtered reads to the human reference genome (GRCh38). Duplicate reads were filtered using Picard-tools-2.1.0 [ 55 ]. Differentially expressed genes were determined using the DEseq package [ 56 ] in R, with cutoff parameters set at fold-change: 2; p < 0.05.…”

Section: Methodsmentioning

confidence: 99%

M. tuberculosis CRISPR/Cas proteins are secreted virulence factors that trigger cellular immune responses

et al. 2021

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The role of prokaryotic CRISPR/Cas system proteins as a defensive shield against invasive nucleic acids has been studied extensively. Non-canonical roles in pathogenesis involving intracellular targeting of certain virulence-associated endogenous mRNA have also been reported for some Type I and Type II CRISPR/Cas proteins, but no such roles have yet been established for Type III system proteins. Here, we demonstrate that M. tuberculosis (Type III-A system) CRISPR/Cas proteins Csm1, Csm3, Csm5, Csm6, and Cas6 are secreted and induce host immune responses. Using cell and animal experiments, we show that Cas6, in particular, provokes IFN-γ release from PBMCs from active tuberculosis (TB) patients, and its deletion markedly attenuates virulence in a murine M. tuberculosis challenge model. Recombinant MTBCas6 induces apoptosis of macrophages and lung fibroblasts, and interacts with the surface of cells in a caspase and TLR-2 independent manner. Transcriptomic and signal pathway studies using THP-1 macrophages stimulated with MTBCas6 indicated that MTBCas6 upregulates expression of genes associated with the NF-κB pathway leading to higher levels of IL-6, IL-1β, and TNF-α release, cytokines known to activate immune system cells in response to M. tuberculosis infection. Our findings suggest that, in addition to their intracellular shielding role, M. tuberculosis CRISPR/Cas proteins have non-canonical extracellular roles, functioning like a virulent sword, and activating host immune responses.

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“…For example, see [148] for Dirichlet fundamental domains in complex hyperbolic spaces. Many results on explicit fundamental domains have been obtained, see the papers [121], [342], [138], [139] [341].…”

Section: Fundamental Domains In Complex Hyperbolic Spacesmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

IRMA Lectures in Mathematics and Theoretical Physics

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SummaryIn this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups Out(F n ), Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of Mod g = Mod g,0 on the Teichmüller space T g to construct an intrinsic fundamental domain consisting of finitely many cells, * Partially Supported by NSF grant DMS 0905283 2 solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne-Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups.The layout of this chapter is as follows. In §1, the introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups Γ of semisimple Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures, and their actions on symmetri...

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Analysis of a Picard modular group

Abstract: Our main goal is to analyze the geometric and spectral properties of the Picard modular group with Gaussian integer entries acting on the two-dimensional complex hyperbolic space.

Cited by 8 publications

References 10 publications

The geometry of the Gauss–Picard modular group

The geometry of the Gauss–Picard modular group

M. tuberculosis CRISPR/Cas proteins are secreted virulence factors that trigger cellular immune responses

A tale of two groups: arithmetic groups and mapping class groups

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