# Maximal Volume Representations Are Fuchsian

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### Explicit bounds on automorphic and canonical Green functions

2.2. Fuchsian groups A Fuchsian group is a discrete subgroup of SL 2(R). A Fuchsian group is co nite if nH has nite volume with respect to the measure induced by H. We will exclusively consider co nite Fuchsian groups, and for such a group we write vol H = Z nH : Let be a co nite Fuchsian group. We de ne the quotient nH in a stack-like way. The

### New Publications Offered by the AMS, Volume 54, Number 2

This volume contains the proceedings of theKorea-Japan Conferenceon Algebraic Geometry in honor of Igor Dolgachev on his sixtieth birthday. The articles in this volume explore a wide variety of problems that illustrate interactions between algebraic geometry and other branches of mathematics. Among the topics covered by this volume are

### Automorphic Representations of SL(2 R) and Quantization of Fields

American Research Journal of Mathematics, Volume 1, Issue 2, April 2015 ISSN 2378-704X www.arjonline.org 28 is the maximal compact subgroup. To each modular form f ∈ Sk(Γ) of weight k on the Poincar´e plane H = SL(2, R)/ SO(2), we associate the

### HITCHIN HARMONIC MAPS ARE IMMERSIONS

representations of ˇ 1() into Grand are deformations of the irreducible Fuchsian representations ˇ 1() ! SL 2R ! Gr which uniformize the surface :For any choice of marked complex structure on the surface and any Hitchin representation, we show that the corresponding equi-variant harmonic map is an immersion. Pulling back the Riemannian

### Geometric structures and representation varieties workshop

For a cusped hyperbolic 3-manifold, one can consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results for classifying the infinite families of hyperbolic 3-manifolds of cusp volume < 2.62 and the implications of this classification. These families are of

### Introduction - Warwick

Fuchsian space of the twice punctured torus. In accordance with a conjecture of the second author, we show that their closures intersect Fuchsian space in the simplices of minima introduced by Kerckho All computations are done using complex Fenchel-Nielsen coordinates for quasi-Fuchsian space referred to a maximal system of curves. 1

### Meromorphic continuation of Selberg zeta functions with

center, K is a maximal twists by inﬁnite-dimensional representations, or for non-Fuchsian spaces of non-ﬁnite volume as well as beyond representations

### Connected components of representation spaces

Representation spaces = nitely generated group G = topological group Hom( ;G)= space of homomorphisms !G. Natural topology as subset of GjSj S a generating set for

### Actes du séminaire de Théorie spectrale et géométrie

Higgs bundles, pseudo-hyperbolic geometry and maximal representations Volume 34 (2016-2017), p. 97-114. Gprovides a natural generalization of Fuchsian

### Arithmetic Fuchsian Groups of Genus Zero

Volume 2, Number 2 (Special Issue: In honor of John H. Coates, Part 2 of 2) 1 31, 2006 Arithmetic Fuchsian Groups of Genus Zero D. D. Long, C. Maclachlan and A. W. Reid 1. Introduction If ¡ is a ﬂnite co-area Fuchsian group acting on H2, then the quotient H2=¡ is a hyperbolic 2-orbifold, with underlying space an orientable surface (possibly

### DRAFTolume representations are fuchsian

Maximal v DRAFT olume representations are fuchsian Stefano Francaviglia University of Pisa, Italy e-mail: [email protected] Ben Kla UQAM, Canada e-mail: kla @math.uic.edu September 22, 2004 Abstract We prove a volume-rigidity theorem for fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(Hn). Namely,

### Teichmüller Theory, Hyperbolicity and Dynamics

gradient flow for renormalized volume Phillip Engel Uniqueness of the measure of maximal entropy for geodesic flows on certain quasi-Fuchsian representations.

### FUCHSIAN SUBGROUPS OF BIANCHI GROUPS

transactions of the

### PARAMETRIZING FUCHSIAN SUBGROUPS OF THE BIANCHI GROUPS

cocompact or non-cocompact. The conjugacy classes of maximal arithmetic Fuchsian subgroups and hence the wide commensurability classes of finite covolume Fuchsian subgroups, can be parametrized by their discriminant (see 3.1) which is a positive in teger related to the circle or straight line stabilized by the Fuchsian group. We prove (3.3).

### Divergent sequences of Kleinian groups

transverse measures and that they are maximal and connected. We can see the assumption of maximality is essential by taking look at an ex-ample of Anderson{Canary [3]. They constructed an example of quasi-Fuchsian groups converging in AH p(S) which correspond to pairs of marked hyperbolic structures (m i;n i) such that fm igand fn

### Introduction - Rice University

examples include the non-uniform lattice of minimal co-volume in PSL(2;C) and the fundamental group of the Weeks manifold (the closed hyperbolic 3{manifold of minimal volume). 1. Introduction It is notoriously hard to unravel the nature of a nitely presented group In order to do so, one must explore how the group can act on di erent kinds of

### The geometry of maximal components of the PSp(4,R) character

The geometry of maximal components of the PSp.4;R/character variety 1253 into a split real Lie group and the space of maximal representations into a Lie group of Hermitian type both deﬁne particularly interesting components of the character variety.

### A complex hyperbolic Riley slice - Durham University

Such a discrete subgroup of SO(2,1) is called R-Fuchsian. A motivating example is the case where Γ is an R-Fuchsian representation ρ0 of π1(Σ), the funda-0 in the representation variety admits a neighbourhood of maximal dimension containing only discrete and faithful representations. See [19, 23, 27].

### archive.ymsc.tsinghua.edu.cn

Geom. Funct. Anal. Vol. 29 (2019) 539 560 https://doi.org/10.1007/s00039-019-00491-7 Published online April 9, 2019 c 2019 Springer Nature Switzerland AG GAFA

### Automorphic Representations of SL(2 ℝ and Quantization of Fields

American Research Journal of Mathematics, Volume 1, Issue 2, April 2015 ISSN 2378-704X www.arjonline.org 26 The representations are uniquely up to equivalence defined by the character, which are defined as some distribution function. Beside of the main goal we make also some clear presentation of automorphic representations in this

### DeroinTholozan DominatingRepresentations IMRN Revisions

Representations with extremal Toledo invariant are called maximal. Surprisingly, maximal representations in that sense do not have maximal length spectrum. Indeed, Toledo s maximal representations are Fuchsian in restriction to some holomorphic copy of H2, which has curvature −4. In constrast, representations that cannot be strictly

### Finite abelian subgroups of the mapping class group

surfaces themselves, with special attention for groups meeting the maximal order bound 84.˙ 1/, the Hurwitz bound. Later in the 1960 s and 1970 s, a more systematic study began, especially using uniformizing Fuchsian groups. For instance, see the papers of Algebraic & Geometric Topology, Volume 7 (2007)

### Fuchsian Subgroups of Bianchi Groups

Volume 348, Number 5, May 1996 FUCHSIAN SUBGROUPS OF BIANCHI GROUPS D. G. JAMES AND C. MACLACHLAN ABSTRACT. A maximal non-elementary Fuchsian subgroup of a Bianchi group PSL(2, Od) has an invariant circle or straight line under its linear fractional action on the complex plane, to which is associated a positive integer D, the

### Divergent sequences of Kleinian groups

transverse measures and that they are maximal and connected. We can see the assumption of maximality is essential by taking look at an ex-ample of Anderson Canary [3]. They constructed an example of quasi-Fuchsian groups converging in AHp(S) which correspond to pairs of marked hyperbolic

### Hermitian symmetric spaces of infinite dimension and maximal

Rigidity for maximal representations Theorem (D.-L ecureux-Pozzetti) Let be a lattice of SU(1;n) with n 1 and ˆ: !PU(p;1) be a maximal representation. If p 2 then there exists a nite dimensional totally geodesic subspace YˆX C(p;1) that is-invariant. More generally, there is no Zariski-dense maximal representation!PU(p;1). 26

### Harmonic maps and representations of non-uniform lattices of

REPRESENTATIONS OF NON-UNIFORM LATTICES OF PU(m,1) 509 volume assumption is needed precisely because lattices in PU(m,1) are not superrigid. More recently, M. Burger and A. Iozzi proved in [3] (see also [20]) that the conjecture is also true for non-uniform lattices of PU(m,1), m> 2, if

### The boundary of the deformation space of the fundamental

The set of all admissible representations modulo conjugation in Conf(S3)is called the deformation space Def(Γ) of the group Γ. The set Def(Γ) inherits the topology of convergence on generators of Γ on com-pact subsets in S 3because Def(Γ) ˆ Conf(S ) k =˘, k2N (˘is conjugation in Conf(S3)). As Def(Γ) is a bounded domain [13] two

### SUPER-MAXIMAL REPRESENTATIONS FROM FUNDAMENTAL GROUPS OF

The volume heavily depends on the conjugacy class of the peripherals formed by a Fuchsian representation j and a super-maximal maximal representations can be

### AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 11, Pages

Apr 24, 2003 Volume 131, Number 11, Pages 3571-3578 S 0002-9939(03)06905-3 Article electronically published on April 24, 2003 THE GROMOV-LAWSON-ROSENBERG CONJECTURE FOR COCOMPACT FUCHSIAN GROUPS JAMES F. DAVIS AND KIMBERLY PEARSON (Communicated by Paul Goerss) ABSTRACT. We prove the Gromov-Lawson-Rosenberg conjecture for cocom-

### GEAR Second Network Retreat, University of Maryland March 17

maximal representations, recovering Mostow rigidity for hyperbolic manifolds. In the cocompact case, the set of values for the volume of a representation is discrete. In even dimension, this follows from the fact that the volume form is an Euler class. In odd dimension, this was proven by Besson, Courtois and Gallot.

### Abstracts - IMPA

(IMPA) Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points Misha Verbitsky (IMPA) Hyperkähler structure on the space of quasi-Fuchsian representations

### Maximal Representations of Surface Groups: Symplectic Anosov

Maximal Representations of Surface Groups 559 We have the inclusion Rep H Γ g,Sp(V) ⊂ Rep max Γ,Sp(V), but while the representations in the Hitchin component are all irreducible [50], there are (at least when dimV ≥ 4) components of maximal representations which contain reducible representations, so that the above inclusion is strict.

### Maximal Representations of Surface Groups: Symplectic Anosov

Maximal Representations of Surface Groups 559 We have the inclusion RepH Γ g,Sp(V) ⊂ Repmax Γ ,Sp(V), but while the representations in the Hitchin component are all irreducible [50], there are (at least when dimV ≥ 4) components of maximal representations which contain reducible representations, so that the above inclusion is strict.

### Dynamics on PSL(2 C -character varieties of certain

In the case that M is a S [0;1]; Goldman conjectured that quasi-Fuchsian space, the interior of AH(S [0;1]) is a maximal domain of discontinuity. In the case that M is a handlebody of genus at least two, Hg; Minsky ([46]) exhibited a domain of discontinuity called the set of primitive-stable representations that is strictly larger than the

### GEAR Junior Retreat { Abstracts

Abstract: Maximal representations are a family of representations of surface groups (i.e. ˇ 1(g)) into the symplectic groups Sp(2n;R) (or, more generally, into Lie groups of Hermitian type). By many aspects, maximal representations generalize uniformization s representations and their moduli spaces generalizes the Teichmuller space.

### MEROMORPHIC CONTINUATION OF SELBERG ZETA FUNCTIONS WITH

Kis a maximal compact subgroup of G, and Γ is a torsion-free cocompact lattice in G) and arbitrary ﬁnite-dimensional representations χ: Γ → GL(V). In addition, he allows unitary twists of K, which we do not discuss here. He establishes an analogue of the Selberg trace formula (which, for unitary representations, is known

### DISSERTATION DOCTEUR DE L UNIVERSITÉ DU LUXEMBOURG EN

Mess parameterised the deformation space of globally hyperbolic maximal anti-de that conjugate two Fuchsian representations. the volume of the convex core of

### The geometry of symplectic quasi-Hitchin representations.

Fuchsian Case quasi-Fuchsian Case One can show that a group that is bi-Lipschitz equivalent to a Fuchsian group is quasi-Fuchsian. Furthermore, BerÕs simultaneous uniformization theorem gives that there is a one-to-one correspondence between quasi-Fuchsian groups considered up to an appropri-

### Maximal representations of surface groups

Maximal Representations of Surface Groups 547 but while the representations in the Hitchin component are all irreducible [50], there are (at least when dimV ≥ 4) components of maximal representations which contain reducible representations, so that the above inclusion is strict.

### Boris N. Apanasov

action of the Fuchsian group ˆ 0 ˆM ob(n 1) ˘=IsomHn. Due to Theorem 1.1 the subvariety R qf() R n() of all quasi-Fuchsian representations is an open connected component of the variety of discrete representations ˆ2 R n(). Equivalently, one can consider a conformal n-manifold (orbifold) M and