# Groups With Domains Of Discontinuity

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### Geometry of compact complex manifolds associated to

groups. In the same way, we propose to generalize the theory of quasi-Fuchsian groups by studying complex deformations of these G-Fuchsian and Hitchin representations and the associated holomorphic actions on parabolic homogeneous spaces of G. The existence of domains of proper discontinuity for such actions follows from a theory

### ABSTRACT GENERALIZATIONS OF SCHOTTKY GROUPS Jean-Philippe

Previous constructions of Schottky groups using hypersurfaces include pro-jective linear groups acting on complex projective spaces ([Nor86], [SV03]), a ne Lorentzian groups acting on R2;1 ([Dru92]), and conformal Lorentzian groups acting on the Einstein universe ([CFLD14], [BCFG17]). This last example is the one we focus on in Chapter3. The

### ANOSOV REPRESENTATIONS: DOMAINS OF DISCONTINUITY AND APPLICATIONS

tations are discrete embeddings of surface groups into PSL(2;R), quasi-Fuchsian representations of surface groups into PSL(2;C), embeddings of free groups as Schottky groups, or embeddings of uniform lattices. Anosov representations into Lie groups of rank one are exactly convex cocompact representations (see Theo-rem1.8)

### Dynamics on ﬂag manifolds: domains of proper discontinuity

domains of proper discontinuity and cocompactness Michael Kapovich, Bernhard Leeb, Joan Porti March 3, 2017 To Guiomar Abstract For noncompact semisimple Lie groups G with ﬁnite center we study the dynamics of the actions of their discrete subgroups Γ ă G on the associated partial ﬂag manifolds G{P.

### )-representation of fundamental groups and related topics

The SL(n)-representation of fundamental groups and related topics Yusuke Inagaki Osaka University Jun 26, 2017 domains of discontinuity and application, Invent.

### Dynamics on flag manifolds: domains of proper discontinuity

Wienhard[11]constructed cocompact domains of proper discontinuity for Anosov subgroups of various semisimple Lie groups acting on various ﬂag manifolds. Cano, Navarrete and Seade[6]extensively studied Kulkarni s limit sets in the case of discrete subgroups of SL.n;C/acting on CPn1.

### A PLY INEQUALITY FOR KLEINIAN GROUPS

The groups with which this paper is concerned do exist. B. Maskit [M1] has used combination theorems to construct Kleinian groups with an element γ of prescribed combinatorial rotation number and a prescribed number of cycles of simply connected domains of discontinuity stabilized by the appropriate iterate of γ.

### The SL(n)-representation of fundamental groups and related topics

N. Hitchin, Lie groups and Teichmuller space, Topology 31 (1992), no.3, 449-473. F. Labourie, Anosov ows, surface groups and curves in projective space, Invent. Math. 165(2006), no. 1, 51-114. Yusuke Inagaki (Osaka Univ.) The SL(n)-representation of fundamental groups and related topics Boston University/Keio University Workshop Jun 29, 2017 14

### Scaling the Root

monolithic commercial registries for large top-level domains such as COM, or DNS-management companies that operate second-level domains with millions of entries. The fact that it is technically feasible to reliably operate very large DNS zones is a necessary, but in no sense sufficient, condition for scaling the root.

### On Boundaries of Teichmuller Spaces and on Kleinian Groups: I

a connected and simply connected region of discontinuity. We call such groups degenerate, their existence has not hitherto been suspected. Although we prove the presence of uncountably many degenerate non-conjugate groups on the boundary of a single T(F), no degenerate group has been thus far constructed explicitly.

### Workshop on Anosov representations: List of talks

Coxeter groups. arXiv:2102.02757. 7 Cocompact domains of discontinuity I This talk will explain how Anosov representations into G = SO(p,q) give rise to cocompact do-mains of discontinuity in spaces of isotropic subspaces of Rp,q, following Frances (case q = 2) and Guichard and Wienhard (general case). This can be used to construct cocompact

### Behavioral Targeting, Machine Learning and Regression

Regression Discontinuity Designs ing methods have enabled targeting of customers in a variety of domains, including pricing, while keeping everything else

### Author's personal copy - Heidelberg University

Author's personal copy C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1057 1060 Geometry/Topology Domains of discontinuity for surface groups Olivier Guicharda,b, Anna Wienhardc a CNRS, laboratoire de mathématiques d Orsay, 91405 Orsay cedex, France

### Indra s Pearls: An Atlas of Kleinian Groups

(3) Tilings of H2, introduction to Fuchsian groups (4) Limit sets and domains of discontinuity (5) Comparison with complex dynamics (6) Schottky groups (7) Maskit s construction of punctured tori (8) Exploration near the boundary of the Maskit slice (9) Methods for drawing pictures (10) Connection with 3-dimensional hyperbolic geometry

### Rational Functions: Intercepts, Asymptotes, and Discontinuity

Discontinuity Reporting Category Functions Topic Exploring asymptotes and discontinuity Primary SOL AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; e) asymptotes.

### Introduction to Quasi-Fuchsian Groups

D0 between domains in C is quasiconformal if it sends small circles 2. (Domain of discontinuity, limit set). so these groups are no longer quasi-Fuchsian.

### arXiv:1306.5832v1 [math.GT] 25 Jun 2013

Wienhard establishing the proper discontinuity of the action of outer automorphism groups on spaces of Anosov representations. 2) The work of Canary, Gelander, Lee, Magid, Minsky and Storm on the case where is the fundamental group of a compact 3-manifold with boundary and G = PSL(2;C). 1. Overview

### The Domain of Solutions To Differential Equations

The reason domains are restricted to intervals will be discussed below, but ﬁrst we present some more examples. Most of these are taken from diﬀerential equations textbooks because, unfortunately, most calculus texts have yet to provide a complete discussion about domains of solutions to initial value problems. See

### DOMAINS OF DISCONTINUITY FOR ALMOST-FUCHSIAN GROUPS

DOMAINS OF DISCONTINUITY FOR ALMOST-FUCHSIAN GROUPS 3 @ 1(H3) into which the limit set of can not penetrate. Lastly in x5.1 we prove the promised theorem: Theorem 1.1. There are no doubly-degenerate geometric limits of almost-Fuchsian groups. There is a technical issue in the proof of the above theorem; in order to estimate the size of the

### A TRACE-CLASS RIGIDITY THEOREM FOR KLEINIAN GROUPS

2 lifts to a diﬀeomorphism of the domains of discontinuity Ω(Γ 1) and Ω(Γ 2) that induces an isomorphism of the groups Γ 1 and Γ 2. This is essentially equivalent to requiring that ψ induce an invertible map from Γ 1-automorphic forms on Ω(Γ 1) to Γ 2-automorphic forms on Ω(Γ 2). We will prove: Theorem 1.1. Suppose that Γ 1 and Γ

### Introduction - Mathematics U-M LSA

nents of domains of discontinuity of Kleinian groups. We recall that a Kleinian group Γ is a discrete subgroup of PSL2(C), regarded as the group of conformal automorphisms of Cb, and that its domain of discontinuity Ω(Γ) is the largest open subset of Cb on which Γ acts properly discontinuously. A Kleinian group Γ is said to be analytically

### Andrew M. Sanders

Domains of Discontinuity for almost-Fuchsian manifolds 1st annual GEAR retreat, UIUC, Urbana-Champaign, Illinois, July 2012. Minimal surfaces in quasi-Fuchsian manifolds and Hausdor dimension. Geometry and Analysis of Surface Groups: special seminar, IHP, Paris, France, February 2012. Hyperbolization of three manifolds and an introduction to

### The Arithmetic of Kleinian Groups

4.Kleinian groups (a)discreteness and discontinuity (Beardon x5.3) (b)basic properties (c)limit sets (d)cusps and stabilizers (e)fundamental domains and geometrical niteness (f)Fuchsian groups 5.Quaternion algebras (Maclachlan and Reid, x2) (a)General theory, hilbert symbols and orders (b)norm form (c)orthogonal groups (d)cli ord algebras

### Handbook of Procedures and Evidence Standards: Version 2

III.6. After attrition, there may be differences between the intervention and comparison groups 51 III.6. After attrition, there may be differences between the intervention and comparison groups 51 III.7. Overall and differential attrition levels that result in high or low attrition 52 III.8.

### Andrew M. Sanders - homepages.math.uic.edu

Domains of Discontinuity for almost-Fuchsian manifolds 1st annual GEAR retreat, UIUC, Urbana-Champaign, Illinois, July 2012. Minimal surfaces in quasi-Fuchsian manifolds and Hausdor dimension. Geometry and Analysis of Surface Groups: special seminar, IHP, Paris, France, February 2012. Hyperbolization of three manifolds and an introduction to

### arXiv:2103.10082v1 [math.GT] 18 Mar 2021

Kleinian groups started with the following. Theorem 2.3 (Klein-Maskit combination theorem for free product with amal-gamation). [Mas65a] Let 1; 2 be Kleinian groups with domains of discontinuity 1; 2 respectively. Let H = 1 2. Let D 1;D 2; be partial fundamental do-mains for 1; 2;Hrespectively. For i = 1,2, set E i= H:D i. Denote the interior

### A fake Schottky group in Mod S

Though G is not a maximal domain of discontinuity, we show in Section 5 that, for the groups in Theorem 1.2, it is nonetheless the intersection of all such maximal domains. 2. Surface dynamics If Xis a subset of PML(S), we let ZX= f[ ] 2PML(S)ji( ; ) = 0 for some [ ] 2Xg be the zero locus of X. If X= f[x]gwe sometimes write Zxfor ZX.

### DOMAINS OF DISCONTINUITY FOR ANOSOV REPRESENTATIONS

2. Automorphism groups of non-degenerate sesquilinear forms We will now describe the construction of domains of discontinuity for a very special type of representation ˆ: !G. Notation 2.1. Let (V;F) be a vector space over either R, C or H, equipped with a non-degenerate quadratic form Fsuch that

### The Classification of Punctured-Torus Groups

the deformation theory of quasi-Fuchsian groups (in any genus) and showed that they are parametrized by a product of Teichmuller spaces (D x D in our case). Maskit [60] further studied the groups that arise on the boundary of these deformation spaces when the domains of discontinuity are pinched and

### Cultural Discontinuities and Schooling

communicative, and interactional domains that are presumed to affect school experience. This paper attempts to refine the cultural discontinuity hypothesis by distinguishing between three types of discontinuities: universal, primary, and secondary discontinuities. It is suggested that each is more or less

### ABSTRACT MINIMAL SURFACES, HYPERBOLIC 3-MANIFOLDS, AND

ABSTRACT Title of dissertation: MINIMAL SURFACES, HYPERBOLIC 3-MANIFOLDS, AND RELATED DEFORMATION SPACES Andrew Sanders, Doctor of Philosophy, 2013 Dissertation directed by: Dr. William Goldman

### Quasiconformal homeomorphisms and dynamics II: Structural

Mar 12, 2021 gate discrete groups with non-trivial domains of discontinuity on (2. Quasi-conformal conjugacies turn up because they are unique on the limit set and so depend holomorphically on the parameter. In w 2 we make the following observation

### What works to enhance women s agency: Cross-cutting lessons

Mar 05, 2020 political domains. Finally, the Additional Notes on Women s Agency section briefly discusses two topics, engaging men and measuring women s agency, that emerged as common themes in the literature but are beyond the scope of this review. In the Crosscutting Findings Section, we summarize how many studies across several domains identified,

### Teacher Rubric 2017-18 Domain 1: Planning and Preparation

groups of students. Teacher actively seeks knowledge of students backgrounds, cultures, skills, language proficiency, interests, and special needs from a variety of sources, and attains this knowledge for individual students. 1c: Setting instructional outcomes* of which permit viable methods of Instructional outcomes are too general, and

### Ghana LEAP 1000 Programme: Baseline Evaluation Report

The discontinuity design successfully generated a valid comparison group to measure programme impacts. We performed over 500 statistical tests for mean (or proportional) differences between the treatment and comparison group across all potential impact domains ranging from consumption and

### Discrete isometry groups of symmetric spaces Michael Kapovich

1. Kleinian groups { discrete isometry groups of negatively curved symmetric spaces 6 2. Geometry of symmetric spaces of noncompact type 16 3. Discrete subgroups: Geometric and dynamical conditions 34 4. Discrete subgroups: Domains of proper discontinuity 47 5. Future directions 68 6. Appendix. Horofunction compacti cation 69 7. Appendix.

### Kleinian Groups and John Domains - DASH Harvard

Kleinian groups and John domains Curtis T. McMullen∗ 5 December, 1996 Abstract We characterize when John domains arise in the setting of Kleinian groups. 1 Introduction A region U in the Riemann sphere is a John domain if every point in U can be reached from a ﬁxed basepoint by a ﬂexible cone with a deﬁnite angle at its vertex.