AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

Quadratic differentials as stability conditions

We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.Comment: 123 pages; 38 figures. Version 2: hypotheses in the main results mildly weakened, to reflect improved results of Labardini-Fragoso and coauthors. Version 3: minor changes to incorporate referees' suggestions. This version to appear in Publ. Math. de l'IHE

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

w-Cycles in Surface Groups

For w an element in the fundamental group of a closed, orientable, hyperbolic surface Ω which is not a proper power, and Σ a surface immersing in Ω, we show that the number of distinct lifts of w to Σ is bounded above by -χ(Σ). In special cases which can be characterised by interdependencies of the lifts of w, we find a stronger bound, whereby the total degree of covering from curves in Σ representing the lifts to the curve representing w is also bounded above by -χ(Σ). This is achieved by a method we introduce for decomposing surfaces into pieces that behave similarly to graphs, and using them to estimate Euler characteristics using a stacking argument of the kind introduced by Louder and Wilton. We then consider some consequences of these bounds for quotients of orientable surface groups by a single element. We demonstrate ways in which these groups behave analogously to one-relator groups; in particular, the ones with torsion are coherent (i.e. all finitely-generated subgroups have finite presentations), and those without torsion possess the related property of non-positive immersions as introduced by Wise

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

The quantum Teichmuller space and its representations

The quantum Teichmüller space is an algebraic object associated with a punctured surface admitting an ideal triangulation. Two somewhat different versions of it have been introduced, as a quantization by deformation of the Teichmüller space of a surface, independently by Chekhov and Fock and by Kashaev. As in the article [1], we follow the exponential version of the Chekhov-Fock approach, whose setting has been established in [2]. In this way, the study is focused on non-commutative algebras and their finite-dimensional representations, instead of Lie algebras and self-adjoint operators on Hilbert spaces, as in [3] and [4]. Given $S$ a surface admitting an ideal triangulation $\lambda$, we can produce a non-commutative \C-algebra $\mathcal{T}^q_\lambda$, generated by variables $X_i^{\pm 1}$ corresponding to the edges of $\lambda$ and endowed with relations $X_i X_j = q^{2 \sigma_{i j}} X_j X_i$, where $\sigma_{i j}$ is an integer number, depending on the mutual position of the edges $\lambda_i$ and $\lambda_j$ in $\lambda$, and q \in \C^* is a complex number. The algebra $\mathcal{T}^q_\lambda$ is called the Chekhov-Fock algebra associated with the surface $S$ and the ideal triangulation $\lambda$. Varying $\lambda$ in the set $\Lambda(S)$ of all the ideal triangulations of $S$, we obtain a collection of algebras, whose fraction rings $\widehat{\mathcal{T}}^q_\lambda$ are related by isomorphisms \mappa{\Phi^q_{\lambda \lambda'}}{\widehat{\mathcal{T}}^q_{\lambda'}}{\widehat{\mathcal{T}}^q_\lambda}. This structure allows us to consider an object realized by "gluing" all the $\widehat{\mathcal{T}}^q_\lambda$ through the maps $\Phi^q_{\lambda \lambda'}$. The result of this procedure is an intrinsic algebraic object, called the quantum Teichmüller space of $S$ and denoted by $\mathcal{T}^q_S$, which does not depend on the chosen ideal triangulation any more. The explicit expressions of the $\Phi_{\lambda \lambda'}^q$ reveal the geometric essence of this algebraic object. These isomorphisms are designed in order to be a non-commutative generalization of the coordinate changes on the ring of rational functions on the classical Teichmüller space $\mathcal{T}(S)$ of a surface $S$ (here $\mathcal{T}(S)$ denotes the space of isotopy classes of complete hyperbolic metrics on $S$). The main purpose of this thesis is the study of the quantum Teichmüller space and the investigation of its finite-dimensional representations. A necessary condition for the existence of a finite-dimensional representation of any Chekhov-Fock algebra $\mathcal{T}^q_\lambda$ is that $q^2$ is a root of unity, hence we always assume that $q^2$ is a primitive $N$-th root of unity, for a certain $N \in \N$. Firstly, we focus on the Chekhov-Fock algebras $\mathcal{T}^q_\lambda$ and the classification of their local and irreducible representations. Then, following [1], we give a notion of finite-dimensional representation of the quantum Teichmüller space, defined as a collection \rho = \set{\rho_\lambda \vcentcolon \mathcal{T}^q_\lambda \rightarrow \End(V_\lambda) }_{\lambda \in \Lambda(S)} of representations of all the Chekhov-Fock algebras associated with the surface $S$, such that $\rho_\lambda \circ \Phi_{\lambda \lambda'}^q$ is isomorphic to $\rho_{\lambda'}'$ for every $\lambda, \lambda' \in \Lambda(S)$. Local (and irreducible) representations of the quantum Teichmüller space turn out to be classified by conjugacy classes of homomorphisms from the fundamental group of the surface $S$ to the group of orientation preserving isometries of the $3$-dimensional hyperbolic space, together with some additional data. In the last part, we study certain linear isomorphisms through which local representations are connected. Given $\rho, \rho'$ two isomorphic local representations of the quantum Teichmüller space of $S$, for every $\lambda, \lambda' \in \Lambda(S)$ the representations $\rho_\lambda \circ \Phi_{\lambda \lambda'}^q$ and $\rho_{\lambda'}'$ are isomorphic through a linear isomorphism $L^{\rho \rho'}_{\lambda \lambda'}$. Such a $L^{\rho \rho'}_{\lambda \lambda'}$ is called an intertwining operator. One of the main purposes of [4] was to select a unique intertwining operator $L^{\rho \rho'}_{\lambda \lambda'}$ for every choice of $\rho, \rho'$ local representations and $\lambda, \lambda'$ ideal triangulations, requiring that the whole system of operators (for varying $\lambda, \lambda' \in \Lambda(S)$, the representations $\rho, \rho'$ and the surface $S$) verifies some natural Fusion and Composition properties, concerning their behaviour with respect to the fusion of representations and changing of triangulations. However, in our investigation of the ideas exposed in [5], we have found a difficulty that compromises the original statement [5,Theorem 20], in particular the possibility to select a unique intertwining operator for every choice of $\rho, \rho', \lambda, \lambda'$. We prove that the best that can be done is the selection of sets of intertwining operators $\mathscr{L}^{\rho \rho'}_{\lambda \lambda'}$, instead of a single linear isomorphism. Each set $\mathscr{L}^{\rho \rho'}_{\lambda \lambda'}$ is endowed with a natural free and transitive action of $H_1(S;\Z_N)$, so its cardinality is always finite, but it goes to $\infty$ by increasing the complexity of the surface $S$ and the number $N \in \N$. In conclusion, we reformulate the theory of invariants for pseudo-Anosov diffeomorphisms developed in [5] in light of these facts. We also expose an explicit calculation of an intertwining operator when $\lambda$ and $\lambda'$ differ by diagonal exchange and $S$ is an ideal square, which is basically the elementary block needed to express a generic intertwining operator. [1] Bonahon, Francis, and Xiaobo Liu. "Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms." Geometry & Topology 11.2 (2007): 889-937. [2] Liu, Xiaobo. "The quantum Teichmüller space as a noncommutative algebraic object." Journal of Knot Theory and its Ramifications 18.05 (2009): 705-726. [3] Fock, Vladimir V., and Leonid O. Chekhov. "A quantum Teichmüller space." Theoretical and Mathematical Physics 120.3 (1999): 1245-1259. [4] Kashaev, Rinat M. "A link invariant from quantum dilogarithm." Modern Physics Letters A 10.19 (1995): 1409-1418. [5] Bai, Hua, Francis Bonahon, and Xiaobo Liu. "Local representations of the quantum Teichmuller space." arXiv preprint arXiv:0707.2151 (2007)