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Proceedings of the London Mathematical Society, ISSN 0024-6115, 06/2019, Volume 118, Issue 6, pp. 1305 - 1327
We investigate the maximal solid tubes around short simple closed geodesics in hyperbolic three‐manifolds and how the complex length of curves relates to... 
57M05 (secondary) | 53A10 | 30F40 (primary)
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 12/2019, Volume 100, Issue 3, pp. 851 - 870
Earlier work introduced a geometrically natural probability measure on the group of all Möbius transformations of the hyperbolic plane so as to be able to... 
30F40 | 30C60 | 53A35 | 30D50 | 22E40 | 57N13 | 20H10 | 57M60 (primary)
Journal Article
Compositio mathematica, ISSN 0010-437X, 06/2019, Volume 155, Issue 6, pp. 1118 - 1170
We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang... 
MATHEMATICS
Journal Article
by Liu, Yi
Journal of Differential Geometry, ISSN 0022-040X, 2019, Volume 111, Issue 3, pp. 457 - 493
In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The... 
Quasi-Fuchsian | Good pants | Homological torsion growth | MATHEMATICS | EQUIDISTRIBUTION | GROWTH | TORSION | HOLONOMY | RANK | HOMOLOGY
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 10/2018, Volume 98, Issue 2, pp. 306 - 328
The overall aim of this note is to initiate a ‘manifold’ theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the... 
11K60 (primary) | 11J83 | 30F40 (secondary) | HAUSDORFF DIMENSION | MATHEMATICS | RATIONAL-POINTS | NUMBERS | GEOMETRICALLY FINITE-GROUPS | SETS | PLANAR CURVES | FUCHSIAN-GROUPS | MANIFOLDS | GEODESICS | VECTORS
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 07/2019, Volume 372, Issue 7, pp. 4977 - 5009
We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated... 
HAUSDORFF DIMENSION | MATHEMATICS | Assouad dimension | regularity dimension | FINITENESS | Patterson-Sullivan measure | Kleinian group | ASSOUAD TYPE DIMENSIONS | ENTROPY
Journal Article
Journal of the Australian Mathematical Society, ISSN 1446-7887, 2019, Volume 108, Issue 1, pp. 1 - 32
We shall explain here an idea to generalize classical complex analytic Kleinian group theory to any odd-dimensional cases. For a certain class of discrete... 
32F10 | 2010 Mathematics subject classification | 32F17 | primary 32J18 | 53C56 | secondary 30F40 | compact non-Kahler manifold | MATHEMATICS | projective structure | Kleinian group theory
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 10/2018, Volume 98, Issue 2, pp. 253 - 274
Marked Schottky space MSg is an explicit model of the quasiconformal deformation space of a Schottky group of rank g, this being isomorphic to the punctured... 
57M50 | 30F60 | 57M60 (secondary) | 32G15 | 30F10 | 30F40 (primary) | 57M10 | FORMS | MATHEMATICS | ABELIAN-VARIETIES | GENERATED KLEINIAN GROUPS | ALGEBRAIC-CURVES | MODULI SPACES | TEICHMULLER-SPACES | Mathematics - Complex Variables
Journal Article
Archiv der Mathematik, ISSN 0003-889X, 1/2019, Volume 112, Issue 1, pp. 13 - 18
Recently, Gareth Jones observed that every finite group G can be realized as the group of automorphisms of some dessin d’enfant $${{\mathcal {D}}}$$ D . In... 
Primary 30F40 | Riemann surfaces | Dessins d’enfants | Mathematics, general | Mathematics | 14H57 | Automorphisms | 11G32 | MATHEMATICS | Dessins d'enfants
Journal Article
Acta Mathematica, ISSN 0001-5962, 9/2015, Volume 215, Issue 1, pp. 55 - 126
We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the $${{\rm... 
Mathematics, general | Mathematics | MATHEMATICS | TREES | REPRESENTATIONS | FOLIATIONS | SPACES | DEGENERATIONS | VALUATIONS | HYPERBOLIC STRUCTURES | CURVES
Journal Article
Geometric and Functional Analysis, ISSN 1016-443X, 10/2012, Volume 22, Issue 5, pp. 1361 - 1399
Let $${\mathbf{delta}_{\mathcal S,\lambda}}$$ denote the Floyd metric on a discrete group G generated by a finite set $${\mathcal S}$$ with respect to the... 
30F40 | Primary 20F65 | 22D05 | Convergence group action | Analysis | Secondary 57M07 | Relatively hyperbolic group | Mathematics | Floyd completion | Locally compact group | 20F67 | MATHEMATICS
Journal Article
Osaka Journal of Mathematics, ISSN 0030-6126, 07/2014, Volume 51, Issue 3, pp. 823 - 828
We give a new proof of the characterization of self-bumping points on the space of Kleinian once-punctured torus groups of Ito [8], based on some recent work... 
MATHEMATICS | DEFORMATION SPACES | LIMITS | COMPLEX | HYPERBOLIC 3-MANIFOLDS | 30F40 | 57M50
Journal Article
Bulletin of the London Mathematical Society, ISSN 0024-6093, 06/2020, Volume 52, Issue 3, pp. 530 - 545
A virtual Schottky group is a Kleinian group K containing a Schottky group G as a finite index normal subgroup. These groups correspond to those groups of... 
30F10 | 20H10 | 30F40 (primary) | MATHEMATICS
Journal Article
Ergodic theory and dynamical systems, ISSN 0143-3857, 02/2018, Volume 38, Issue 1, pp. 336 - 370
Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr. 7 (2004), 69–100 (electronic)] introduced the... 
MATHEMATICS | MATHEMATICS, APPLIED | ENERGY | ANOSOV | DIMENSION | CURVATURE | BRANCHED-IMMERSIONS | EXTENSION | POINTS | TOPOLOGICAL-ENTROPY | BIFURCATION | Lower bounds | Riemann manifold | Theorems | Riemann surfaces | Minimal surfaces | Entropy | Rigidity | Manifolds (mathematics)
Journal Article
Geometric and Functional Analysis, ISSN 1016-443X, 11/2010, Volume 20, Issue 5, pp. 1210 - 1230
In this paper we describe a function F n : R + → R + such that for any hyperbolic n-manifold M with totally geodesic boundary $${\partial M \neq \emptyset}$$ ,... 
30F40 | Hyperbolic manifold | Analysis | 30F60 | geodesic boundary | Mathematics | 37D35 | MATHEMATICS
Journal Article
Proceedings of the London Mathematical Society, ISSN 0024-6115, 08/2017, Volume 115, Issue 2, pp. 411 - 447
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question,... 
30F40 | 58J53 | 57M50 | 53C22 | ISOSPECTRAL MANIFOLDS | COMMENSURABILITY CLASSES | SPACES | KLEINIAN-GROUPS | DEHN SURGERY | KNOTS | LINK COMPLEMENTS | MATHEMATICS | MUTATION | VOLUME HYPERBOLIC LATTICES | GEOMETRY
Journal Article
Proceedings of the American Mathematical Society, ISSN 0002-9939, 11/2017, Volume 145, Issue 11, pp. 4703 - 4715
We describe a minimal global coordinate system of order 30 on the \textup {SL}(4,\mathbb{C})-character variety of a rank 2 free group. Using symmetry within... 
Complex hyperbolic space | Two-generator subgroup | Character variety | Traces | 4 × 4 invariants | TOPOLOGY | MATHEMATICS, APPLIED | X 4 MATRICES | GENERATORS | SPACES | traces | 4 x 4 invariants | complex hyperbolic space | MATHEMATICS | ALGEBRAS | GROUP-CHARACTER VARIETIES | two-generator subgroup
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 09/2009, Volume 149, Issue 3, pp. 411 - 460
In this article, we pursue the study of the holomorphic dynamics of mapping class groups on two-dimensional character varieties, also called trace-map dynamics... 
MATHEMATICS | MAXIMAL ENTROPY | LAMINAR CURRENTS | MODULAR GROUP ACTION | DIMENSION | POLYNOMIAL DIFFEOMORPHISMS | DYNAMICS | C-2 | AUTOMORPHISMS | ERGODIC-THEORY | TRACE MAP | 30F40 | 37F | 47B39 | 57M | 34M | 37A | 47B | 37C
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 02/2010, Volume 151, Issue 2, pp. 279 - 336
Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary We prove that there are upper and lower bounds on the diameter of the skinning map of... 
3-MANIFOLD GROUPS | MATHEMATICS | TREES | DEGENERATIONS | THEOREM | RIGIDITY | KLEINIAN-GROUPS | TOTALLY GEODESIC BOUNDARIES | DEFORMATION | HYPERBOLIC CONE-MANIFOLDS | Mathematics - Geometric Topology | 30F40 | 30F60 | 57M50
Journal Article
Geometriae Dedicata, ISSN 0046-5755, 2/2016, Volume 180, Issue 1, pp. 117 - 134
Let $$M=H_1\cup _S H_2$$ M = H 1 ∪ S H 2 be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider... 
Geometry | 30F40 | 57M07 | Mathematics | Heegaard splitting | Projective lamination space | Primary 57M50 | 20F34 | Mapping class group | MATHEMATICS | CURVE COMPLEX | 2-BRIDGE SPHERES | AUTOMORPHISMS | SIMPLE LOOPS | GEOMETRY | Laminated materials
Journal Article
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