scholarly journals Deforming the R-Fuchsian (4, 4, 4)-triangle group into a lattice

Topology ◽  
2006 ◽  
Vol 45 (6) ◽  
pp. 989-1020 ◽  
Author(s):  
Martin Deraux
Keyword(s):  
Triangle Group

scholarly journals The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

scholarly journals Introducing edge-biregular maps

2021 ◽  
Author(s):  
Olivia Reade
Keyword(s):  
Automorphism Group ◽  
Euler Characteristic ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Triangle Group ◽  
Supporting Surface ◽  
Regular Maps ◽  
Special Cases ◽  
Edge Colourings

AbstractWe introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of $$T_{k,l}$$ T k , l , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

scholarly journals A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

2018 ◽  
Vol 40 (3) ◽  
pp. 612-662
Author(s):  
ALEXANDER ADAM ◽  
ANKE POHL
Keyword(s):  
Transfer Operator ◽  
Zeta Functions ◽  
Cusp Forms ◽  
Dimensional Representation ◽  
Triangle Group ◽  
Selberg Zeta Function ◽  
One Dimensional ◽  
Transfer Operators ◽  
Finite Dimensional ◽  
Automorphic Functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

On the Subgroups of the Triangle Group *2p∞

2011 ◽  
Author(s):  
E. D. B. Provido ◽  
M. L. A. N. de las Penas ◽  
M. C. B. Decena
Keyword(s):  
Triangle Group

Quadrilateral cell graphs of the normalizer with signature (2,4,∞)

2020 ◽  
Vol 57 (3) ◽  
pp. 408-425
Author(s):  
Nazli Yazici Gözütok ◽  
Bahadir Özgür Güler
Keyword(s):  
Triangle Group ◽  
Suborbital Graphs ◽  
Imprimitive Action

AbstractIn this study, we investigate suborbital graphs Gu,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group (N) and then obtain some properties of the graphs arising from this action.

scholarly journals Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

2011 ◽  
Vol 33 (1) ◽  
pp. 247-283 ◽  
Author(s):  
M. MÖLLER ◽  
A. D. POHL
Keyword(s):  
Zeta Function ◽  
Symbolic Dynamics ◽  
Fredholm Determinant ◽  
Transfer Operator ◽  
Orbit Space ◽  
Cusp Forms ◽  
Operator Family ◽  
Triangle Group ◽  
Selberg Zeta Function ◽  
Triangle Groups

AbstractWe characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover, we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

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