On groups generated by two positive multi-twists: Teichmueller curves and Lehmer's number

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. In addition, we describe a connection to work of McMullen on Coxeter groups and related work of Hironaka on a construction of an interesting class of fibered links.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper36.abs.htm

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈J(ℚ) is positive semidefinite, where CJ∈J(ℤ) is the incidence matrix of J. Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number  cJ. In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number   čJ, and the defect homomorphism ∂J:ℤJ→ℤ. In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity. In case GJ is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets J with the Coxeter spectral properties of a simply laced Euclidean diagram DJ∈{̃n,̃6,̃7,̃8} associated with J. Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ,  čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix A∈n(ℤ), there is B∈n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix

Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C#

We present combinatorial algorithms constructing loop-free P-critical edge-bipartite (signed) graphs Δ′, with n ≥ 3 vertices, from pairs (Δ, w), with Δ a positive edge-bipartite graph having n-1 vertices and w a sincere root of Δ, up to an action ∗ : UBigrn × O(n, Z) → UBigrn of the orthogonal group O(n, Z) on the set UBigrn of loop-free edge-bipartite graphs, with n ≥ 3 vertices. Here Z is the ring of integers. We also present a package of algorithms for a Coxeter spectral analysis of graphs in UBigrn and for computing the O(n, Z)-orbits of P-critical graphs Δ in UBigrn as well as the positive ones. By applying the package, symbolic computations in Maple and numerical computations in C#, we compute P-critical graphs in UBigrn and connected positive graphs in UBigrn, together with their Coxeter polynomials, reduced Coxeter numbers, and the O(n, Z)-orbits, for n ≤ 10. The computational results are presented in tables of Section 5

Noncommutative geometry on trees and buildings

We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum

Algorytmy numeryczne w spektralnej analizie Coxetera bigrafów.

Celem rozprawy jest rozwiązanie klasy problemów algorytmiczno-obliczeniowych (sformułowanych w pracach [48]–[50], [69]–[76]) występujących w spektralnej klasyfikacji Coxetera-Grama spójnych dodatnich prostych grafów oznakowanych, a także spójnych dodatnich krawędziowo-dwudzielnych grafów ∆ bez pętli (w skrócie bigrafów) zdefiniowanych w [70]. Rozdziały 1 oraz 2 poświęcone są krótkiemu wprowadzeniu do spektralnej analizy Coxetera grafów oznakowanych oraz klasy nieujemnych grafów krawędziowo-dwudzielnych ∆ bez pętli a także podaniu motywacji do badań nad problemami spektralnej klasyfikacji Coxetera skończonych bigrafów. Przypomnijmy, że z dowolnym bigrafem ∆ bez pętli o skończonym zbiorze ponumerowanych n >= 1 wierzchołków, stowarzysza się jego zespolone spektrum Coxetera specc_∆ ⊆ C, tj. spektrum Z-odwracalnej macierzy Coxetera Cox_∆ := −Ǧ_∆ · Ǧ_∆^{−tr} ∈ Mn(Z), gdzie Ǧ_∆ ∈ Mn(Z) jest niesymetryczną macierzą Grama bigrafu ∆. Jednym z celów rozprawy jest podanie klasyfikacji dodatnich grafów krawędziowo-dwudzielnych z dokładnością do silnej Z-kongruencji Grama, zdefiniowanej w pracy [70] następująco: “∆ ≈Z ∆' ⇐⇒ Ǧ_∆' = B^tr · Ǧ_∆ · B, dla pewnej macierzy B ∈ Gl(n, Z)”. Innym z celów rozprawy jest zbudowanie algorytmów symbolicznych i numerycznych obliczających macierz B ∈ Gl(n, Z) definiującą Z-kongruencję Grama ∆ ≈Z ∆', dla dowolnej pary bigrafów spełniających relację ∆ ≈Z ∆' (lub takich, dla których zachodzi równość specc_∆ = specc_∆' ich spektrów Coxetera). W rozdziale 3 przedstawiamy narzędzia techniczne i algorytmiczne pozwalające zredukować rozważane problemy spektralnej klasyfikacji Coxetera-Grama do badania analogicznych problemów dla pewnego skończonego zbioru Mor_D ⊆ Mn(Z) wszystkich morsyfikacji macierzowych A (zdefiniowanego w pracach [69]–[71]) dla ustalonego jednorodnego diagramu Dynkina D ∈ {An, n >= 1, Dn, n ­>= 4, E6, E7, E8}. Zbiór Mor_D jest niezmienniczym podzbiorem zbioru macierzy Mn(Z) względem działania ∗ : Mn(Z) × Gl(n, Z)_D → Mn(Z), (A, B) |→ A ∗ B := B^tr · A · B ograniczonego do skończonej podgrupy Gl(n, Z)_D grupy liniowej Gl(n, Z), zwanej grupą izotropii diagramu D (zobacz [70]). Jednym z ważniejszych wyników tego rozdziału są autorskie algorytmy obliczające zbiór Mor_D, grupę izotropii Gl(n, Z)_D oraz zbiór Gl(n, Z)_D -orbit w Mor_D, dla dowolnego jednorodnego diagramu Dynkina D, a także wyniki tych obliczeń. Jednym z głównych osiągnięć tej rozprawy jest przedstawiona w rozdziale 4 pełna klasyfikacja spójnych dodatnich grafów krawędziowo-dwudzielnych ∆ o co najwyżej 9-ciu wierzchołkach, z dokładnością do silnej Z-kongruencji Grama ∆ ≈Z ∆'. Udowodnione twierdzenie klasyfikacyjne orzeka, że każdy taki bigraf jest Z-kongruentny z jednym z 26 bigrafów klasyfikujących. Rozdział 5 zawiera konstrukcje nieskończonych serii morsyfikacji A ∈ Mor_An diagramu Dynkina An, o parami różnych spektrach Coxetera, dla n >= 1. W rozdziałach 6 oraz 7 przedstawiamy ideę redukcji badanych problemów klasyfikacyjnych do konstrukcji klasy symbolicznych algorytmów toroidalno-oczkowych konstruujących macierze B definiujące silną Z-kongruencję Grama ∆ ≈Z ∆'. W szczególności, dla klasy nieskończonych serii morsyfikacji diagramu Dynkina D = An budujemy algorytmy symboliczne konstruujące, dla dowolnej pary morsyfikacji A, A' ∈ Mor_D leżących w tej samej Gl(n, Z)_D-orbicie, pewną macierz B ∈ Gl(n, Z) taką, że A' = A ∗ B^tr. W drugiej uzupełnionej wersji rozprawy dodaliśmy rozdział 8, w którym szacujemy złożoność obliczeniową stosowanych algorytmów. Główne wyniki rozprawy zostały opublikowane w artykułach [8], [9], [24], [27]–[31].The aim of this thesis is to solve class of the algorithmic and computing problems (formulated in [48]–[50], [69]–[76]) for Coxeter-Gram spectral classification of the connected positive simple signed graphs and connected positive loop-free edge-bipartite graphs ∆ (bigraphs, in short) defined in [70]. The first and the second chapter are devoted to short introduction to the Coxeter spectral analysis of the signed graphs and the class of the loop-free non-negative edge-bipartite graphs ∆. Moreover, we present a motivation for the study of Coxeter spectral analysis problems of the finite edge-bipartite graphs. Recall that, with any loop-free bigraph ∆ with n >= 1 vertices numbered by the integers, we associate the (complex) Coxetera spectrum specc_∆ ⊆ C, i.e., the spectrum of the Z-invertible Coxeter matrix Cox_∆ := −Ǧ_∆ · Ǧ_∆^{−tr} ∈ Mn(Z), where Ǧ_∆ ∈ Mn (Z) is the non-symmetric Gram matrix of ∆. One of the aims of this thesis is to classify positive edge-bipartite graphs, up to the Gram Z-bilinear congruence defined in [70] as follows “∆ ≈Z ∆' ⇐⇒ Ǧ_∆' = B^tr · Ǧ_∆ · B, for some B ∈ Gl(n, Z)”. Another issue of this thesis is to construct symbolic and numeric algorithms for computing matrix B ∈ Gl(n, Z) defining the Gram Z-bilinear congruence ∆ ≈Z ∆', for any pair of edge-bipartite graphs such that ∆ ≈Z ∆' (or their Coxeter spectra equality holds specc_∆ = specc_∆'). In the third chapter we present technical and algorithmic tools that allows a reduction of the Coxeter-Gram spectral classification problems to the analogue problems for some finite set Mor_D ⊆ Mn(Z) of matrix morsifications A (defined in [69]–[71]) of the fixed simply laced Dynkin diagram D ∈ {An, n >= 1, Dn, n >= 4, E6, E7, E8}. Set Mor_D is invariant subset of Mn(Z) under the action ∗ : Mn(Z)×Gl(n, Z)_D → Mn(Z), (A, B) |→ A∗B := B^tr ·A·B limited to isotropy group Gl(n, Z)_D of the diagram D, that is a finite subgroup of the general linear group Gl(n, Z) (see [70]). Main results of this chapter are our algorithms computing the set MorD, isotropy group Gl(n, Z)_D and the set of Gl(n, Z)_D-orbits in Mor_D, for any simply laced Dynkin diagram D. We also present results of these computer calculations. One of the main results of this thesis is a complete classification of positive connected edge-bipartite graphs with at most nine vertices, up to the Gram Z-bilinear congruence ∆ ≈Z ∆', presented in the fourth chapter. It follows from our classification theorem that any of those bigraphs are Z-congruent with one of the 26 classifying bigraphs presented in chapter four. The fifth chapter contains constructions of several infinite series of matrix morsification A ∈ Mor_An for the diagram An, with pairwise different Coxeter spectra, for n >= 1. In the six and the seventh chapter we present an idea of a reduction from studied classification problems to the build class of symbolic toroidal mesh algorithms for computing matrix B ∈ Gl(n, Z) defining the Gram Z-bilinear congruence ∆ ≈Z ∆'. In particular, for a class of the infinite series of matrix morsification for Dynkin diagram D = An we build symbolic algorithms constructing some matrix B ∈ Gl(n, Z) such that A' = A ∗ B^tr , for any pair of morsifications A, A' ∈ Mor_D, that lie in the same Gl(n, Z)D -orbit. In the second supplemented version of this thesis we added the eighth chapter in which we estimate the complexity of used algorithms. The main results of this thesis have been published in the articles [8], [9], [24], [27]–[31]

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions