ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 05 Oct 2022 15:46:22 GMT2022-10-05T15:46:22Z50121- Difference sets in dihedral groupshttps://scholarbank.nus.edu.sg/handle/10635/103133Title: Difference sets in dihedral groups
Authors: Leung, K.H.; Ma, S.L.; Wong, Y.L.
Abstract: We study the existence of nontrivial (2 m, k, λ)-difference sets in dihedral groups. Some nonexistence results are proved. In particular, we show that n = k - λ is odd and φ{symbol}(n)/n < 1/2. Finally, a computer search shows that, except 5 undecided cases, no nontrivial difference set exists in dihedral groups for n ≤ 106. © 1992 Kluwer Academic Publishers.
Sun, 01 Dec 1991 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031331991-12-01T00:00:00Z
- Acyclic groups and wild arcshttps://scholarbank.nus.edu.sg/handle/10635/104688Title: Acyclic groups and wild arcs
Authors: Berrick, A.J.; Wong, Y.-L.
Abstract: We discuss two classes of acyclic groups that are commutator subgroups of finitely presented groups with infinite cyclic abelianization. The first is algebraic and includes groups first exhibited by Baumslag and Gruenberg, of which it is shown that Epstein's acyclic group is a special case. The second class is geometric, and is shown to include a number of wild arc groups in the literature.
Wed, 01 Dec 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046882004-12-01T00:00:00Z
- Generalizations of McShane's identity to hyperbolic cone-surfaceshttps://scholarbank.nus.edu.sg/handle/10635/103324Title: Generalizations of McShane's identity to hyperbolic cone-surfaces
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface [19] to a general identity for hyperbolic cone-surfaces (with all cone angles ≤ π), possibly with cusps and/or geodesic boundary. The general identity is obtained by studying gaps formed by simple-normal geodesies emanating from a distinguished cone point, cusp or boundary geodesic. In particular, by applying the generalized identity to the quotient orbifolds of a hyperbolic one-cone/one-hole torus by its elliptic involution and of a hyperbolic closed genus two surface by its hyperelliptic involution, we obtain general Weierstrass identities for the one-cone/one-hole torus, and an identity for the genus two surface, which are also obtained by McShane using different methods in [20], [22] and [21]. We also give an interpretation of the general identity in terms of complex lengths of the cone points, cusps and geodesic boundary components.
Sun, 01 Jan 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033242006-01-01T00:00:00Z
- On quotients of braid groupshttps://scholarbank.nus.edu.sg/handle/10635/103745Title: On quotients of braid groups
Authors: Leung, K.H.; Wong, Y.L.
Abstract: Let G be the group ℤ[t, t-1] ⋊ ℤ. By studying the action of the braid group Bn on the set Gn, we obtain representations of Bn into a wreath product of the symmetric group and the general linear group over ℤ[t, t-1]. This in particular recovers the Burau representation of the braid group. Furthermore, some quotients of the braid group are obtained by using the representations found. Copyright © 1996 by Marcel Dekker, Inc.
Mon, 01 Jan 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037451996-01-01T00:00:00Z
- Necessary and sufficient conditions for Mcshane's identity and variationshttps://scholarbank.nus.edu.sg/handle/10635/103596Title: Necessary and sufficient conditions for Mcshane's identity and variations
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γ to SL(2,C), this was further generalized by the authors to obtain sufficient conditions for a generalized McShane's identity to hold for arbitrary (not necessarily type-preserving) non-reducible representations in Tan et al. (Submitted). Here we extend the above by giving necessary and sufficient conditions for the generalized McShane identity to hold (Akiyoshi, Miyachi and Sakuma had proved it for type-preserving representations). We also give a version of Bowditch's variation of McShane's identity to once-punctured torus bundles, in the case where the monodromy is generated by a reducible element, and provide necessary and sufficient conditions for the variations to hold.
Sat, 01 Apr 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035962006-04-01T00:00:00Z
- Generalized Markoff maps and McShane's identityhttps://scholarbank.nus.edu.sg/handle/10635/103330Title: Generalized Markoff maps and McShane's identity
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: We study the (relative) SL (2, C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds. © 2007 Elsevier Inc. All rights reserved.
Wed, 30 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033302008-01-30T00:00:00Z
- Delambre-Gauss formulas for augmented, right-angled hexagons in hyperbolic 4-spacehttps://scholarbank.nus.edu.sg/handle/10635/103119Title: Delambre-Gauss formulas for augmented, right-angled hexagons in hyperbolic 4-space
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: We study the geometry of right-angled hexagons in the hyperbolic 4-space H 4 via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon and arbitrarily orient each line and plane involved, so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angles associated to a certain orientation-preserving isometry of the Euclidean 3-space. We also define appropriate complex half side-lengths for the augmented sides of the augmented hexagon. We further explain how to geometrically read off the quaternion half side-lengths for a given oriented, augmented, right-angled hexagon in H 4. Our main result is a set of generalized Delambre-Gauss formulas for oriented, augmented, right-angled hexagons in H 4, involving the quaternion half side-lengths and the complex half side-lengths. We also show in the appendix how the same method gives Delambre-Gauss formulas for oriented right-angled hexagons in H 3, from which the well-known laws of sines and of cosines can be deduced. These formulas generalize the classical Delambre-Gauss formulas for spherical/hyperbolic triangles. © 2012 Elsevier Ltd.
Wed, 20 Jun 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031192012-06-20T00:00:00Z
- Configurations, braids, and homotopy groupshttps://scholarbank.nus.edu.sg/handle/10635/103037Title: Configurations, braids, and homotopy groups
Authors: Berrick, A.J.; Cohen, F.R.; Wong, Y.L.; Wu, J.
Sat, 01 Apr 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030372006-04-01T00:00:00Z
- Corrigendum to "Generalized Markoff maps and McShane's identity" [Adv. Math. 217 (2008) 761-813] (DOI:10.1016/j.aim.2007.09.004)https://scholarbank.nus.edu.sg/handle/10635/104677Title: Corrigendum to "Generalized Markoff maps and McShane's identity" [Adv. Math. 217 (2008) 761-813] (DOI:10.1016/j.aim.2007.09.004)
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Sun, 20 Dec 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046772009-12-20T00:00:00Z
- The SL(2, ℂ) character variety of a one-holed torushttps://scholarbank.nus.edu.sg/handle/10635/104355Title: The SL(2, ℂ) character variety of a one-holed torus
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: In this note we announce several results concerning the SL(2, ℂ) character variety χ of a one-holed torus. We give a description of the largest open subset χBQ of χ on which the mapping class group Γ acts properly discontinuously, in terms of two very simple conditions, and show that a series identity generalizing McShane's identity for the punctured torus holds for all characters in this subset. We also give variations of the McShane-Bowditch identities for characters fixed by an Anosov element of Γ with applications to closed hyperbolic three-manifolds. Finally we give a definition of end invariants for SL(2, ℂ) characters and give a partial classification of the set of end invariants of a character in χ. © 2005 American Mathematical Society.
Sat, 01 Jan 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043552005-01-01T00:00:00Z
- End invariants for SL(2,ℂ) characters of the one-holed torushttps://scholarbank.nus.edu.sg/handle/10635/103199Title: End invariants for SL(2,ℂ) characters of the one-holed torus
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: We define and study the set ε(p) of end invariants of an SL(2, ℂ) character p of the one-holed torus T. We show that the set ε(p) is the entire projective lamination space ℘ℒ of T if and only if p corresponds to the dihedral representation or p is real and corresponds to an SU(2) representation; and that otherwise, ε(p) is closed and has empty interior in ℘ℒ. For real characters p, we give a complete classification of ε(p), and show that ε(p) has either 0, 1 or infinitely many elements, and in the last case, ε(p) is either a Cantor subset of ℘ℒ or is ℘ℒ itself. We also give a similar classification for "imaginary" characters where the trace of the commutator is less than 2. Finally, we show that for characters with discrete simple length spectrum (not corresponding to dihedral or SU(2) representations), ε(p) is a Cantor subset of ℘ℒ if it contains at least three elements. © 2008 by The Johns Hopkins University Press.
Tue, 01 Apr 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031992008-04-01T00:00:00Z
- McShane's identity for classical Schottky groupshttps://scholarbank.nus.edu.sg/handle/10635/103539Title: McShane's identity for classical Schottky groups
Authors: Tan, S.P.; Wong, Y.L.; Zhang, Y.
Abstract: In 1998, Greg McShane demonstrated a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. In 2006, we generalized this to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical Schottky groups. As a consequence, we obtain some surprising new identities in the case of Fuchsian Schottky groups. For classical Schottky groups of rank 2, we also give generalizations of the Weierstrass identities, given by McShane in 2004.
Mon, 01 Sep 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035392008-09-01T00:00:00Z