Automorphism groups of compact bordered Klein surfaces with invariant subsets

In this work we get upper bounds for the order of a group of automorphisms of a compact bordered Klein surface S of algebraic genus greater than 1. These bounds depend on the algebraic genus of S and on the cardinals of finite subsets of S which are invariant under the action of the group. We use our results to obtain upper bounds for the order of a group of automorphism whose action on the set of connected components of the boundary of S is not transitive. The bounds obtained this way depend only on the algebraic genus of S. The author is partially supported by the European Network RAAG HPRN-CT-2001-00271 and the Spanish GAAR DGICYT BFM2002-04797.     

On tangentially transitive projective planes

The existence of Baer collineations in a projective plane is related to the existence of desargues-like configurations. The plane of order four is characterized as the only finite plane that possesses a Baer subplane partition into tangentially transitive Baer subplanes which is preserved by each of the tangentially transitive groups. It is shown that a finite projective plane has either no or one tangentially transitive Baer subplane or is partially transitive of Hughes type (4, m), (5, m) or (6, m) for some m. The Lenz-Barlotti classes which contain a finite plane which is not a translation plane nor its dual and which possesses a tangentially transitive Baer subplane are shown to be classes I.1 and II.1.     

Linear systems with removable base loci

Abstract

We develop several local approaches for the three classes of finite groups: T-groups (normality is a transitive relation) and PT-groups (permutability is a transitive relation) and PST-groups (S-permutability is a transitive relation). Here a subgroup of a finite group G is S-permutable if it permutes with all the Sylow subgroup of G.     

Global Dimension of Transitive Square-Free Rings

Each transitive square-free ring R has a canonical poset. In this paper we investigate connections between the structure of this poset and the projective dimensions of the simple R-modules. We then use these results to characterize the posets of transitive square-free rings with global dimension less than 3.     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type A₂, simple transitive 2-representations are classified by cell 2-representations. In the general case we conjecture that there exist many simple transitive 2-representations which are not cell 2-representations and provide some evidence for our conjecture.     

Topological groups and recurrence of quasi transitive graphs

We present all steps which are necessary in order to classify all locally finite, infinite graphs which carry a quasi transitive random walk that is recurrent. Some new and/or simpler proofs are given. Most of them rely on the fact that autmomorphism groups of locally finite graphs are locally compact with respect to the topology of pointwise convergence—this allows the use of integration on these groups. Conferenza tenuta il 28 novembre 1994     

Quasi-isometries between graphs and trees

Criteria for quasi-isometry between trees and general graphs as well as for quasi-isometries between metrically almost transitive graphs and trees are found. Thereby we use different concepts of thickness for graphs, ends and end spaces. A metrically almost transitive graph is quasi-isometric to a tree if and only if it has only thin metric ends (in the sense of Definition 3.6). If a graph is quasi-isometric to a tree then there is a one-to-one correspondence between the metric ends and those d-fibers which contain a quasi-geodesic. The graphs considered in this paper are not necessarily locally finite.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Planar embeddings with infinite faces

We investigate vertex‐transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r‐valent tree with K₂ are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r = 3 that no vertex‐transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex‐transitive graphs of connectivity 3 and an infinite family of 4‐connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 257–275, 2003     

Finite Homogeneous 3-Graphs

By “3-graph” we mean a pair (V, E) such that E ? [V]³. We show that the only non-trivial finite 3-graphs homogeneous in the sense of Fraïssé are those associated with the projective planes over GF(2) and GF(3), and with the projective lines over GF(5) and GF(9). To exclude other possibilities we use the classification of doubly transitive finite permutation groups.     

Realizing finite edge‐transitive orientable maps

J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) ( 5 ) established that the automorphism group of an edge‐transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge‐transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3,11 (mod 12), there exist finite, orientable, edge‐transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge‐transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge‐transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge‐transitive map, namely, as ?_n × ?₂ where n ≡ 2 (mod 4). © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 1–34, 2001     

Maximal transitive sets with singularities for genericC 1 vector fields

A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC ¹ generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC ¹ vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.     

Automorphism Groups of Trees Acting Locally with Affine Permutations

We consider the structure of groups that act on a p ⁿ -regular tree in a vertex transitive way with the local action (i.e. the action of the vertex stabilizer on the link) isomorphic to the group of affine transformations on a finite affine line.     

On the galois number and the minimal degree of doubly transitive groups

Consider the action of a finite group G on a set M. Then the Galois number is defined to be 1 + f, where fis the maximal number of fixed points of an element in G, which does not act as the identity on M. We determine the Galois number and the minimal degree of all doubly transitive permutation groups.     

Finite permutation groups containing a transitive abelian subgroup

In this paper, we classify finite permutation groups with a transitive abelian subgroup that are almost simple, quasiprimitive and innately transitive, which extend the results of Li and Praeger that is on finite permutation groups with a transitive cyclic subgroup.     

The dynamics of open quantum systems: accessibility results

In this paper we study the accessibility properties of finite dimensional (N-level) open quantum systems in the presence of dissipation and relaxation described by the Lindblad master equation. We specifically focus on the unital Lindbladian case where general results can be obtained. The theory of transitive Lie-group actions is used to classify the system Lie-algebras of the Lindblad equation for which the reachable sets have nonempty interior. For the special case of n -coupled spin-1/2 systems, we obtain a particularly simple characterization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amenable actions of nonamenable groups

We present two methods of constructing amenable (in the sense of Greenleaf) actions of nonamenable groups. In the first part of the paper, we construct a class of faithful transitive amenable actions of the free group using Schreier graphs. In the second part, we show that every finitely generated residually finite group can be embedded into a bigger residually finite group, which acts level-transitively on a locally finite rooted tree, so that the induced action on the boundary of the tree is amenable on every orbit. Bibliography: 25 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 85–96.     

The fluted fragment with transitive relations

The fluted fragment is a fragment of first-order logic (without equality) in which, roughly speaking, the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that this fragment has the finite model property. We consider extensions of the fluted fragment with various numbers of transitive relations, as well as the equality predicate. In the presence of one transitive relation (together with equality), the finite model property is lost; nevertheless, we show that the satisfiability and finite satisfiability problems for this extension remain decidable. We also show that the corresponding problems in the presence of two transitive relations (with equality) or three transitive relations (without equality) are undecidable, even for the two-variable sub-fragment.