Alice Devillers

On the simple connectedness of certain subsets of buildings

Abstract We prove a rank 3 criterion for the simple connectedness of certain subsets of buildings and we give two applications of this criterion. The first generalizes a result of Tits for Chevalley groups to 3-spherical Kac-Moody groups. The second is the proof of the simple connectedness of certain flipflop geometries introduced in [Bennett C., Gramlich R., Hoffman C., Shpectorov S.: Curtis-Phan-Tits theory. In: Groups, combinatorics and geometry (Durham, 2001). World Sci. Publishing, River Edge, NJ, 2003, 13–29].

Line graphs and geodesic transitivity

For a graph Γ, a positive integer s and a subgroup G ≤ Aut(Γ), we prove that G is transitive on the set of s -arcs of Γ if and only if Γ has girth at least 2( s − 1) and G is transitive on the set of ( s − 1)-geodesics of its line graph. As applications, we first classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive. Secondly we prove that the only non-complete locally cyclic 2-geodesic transitive graphs are the octahedron and the icosahedron.

Locally s-distance transitive graphs

We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s � 2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.

ON IMPRIMITIVE RANK 3 PERMUTATION GROUPS

A classification is given of rank 3 group actions which are quasiprim- itive but not primitive. There are two infinite families and a finite number of individual imprimitive examples. When combined with earlier work of Bannai, Kantor, Liebler, Liebeck and Saxl, this yields a classification of all quasiprimitive rank 3 permutation groups. Our classification is achieved by first classifying im- primitive almost simple permutation groups which induce a 2-transitive action on a block system and for which a block stabiliser acts 2-transitively on the block. We also determine those imprimitive rank 3 permutation groups G such that the induced action on a block is almost simple and G does not contain the full socle of the natural wreath product in which G embeds.

Rank 3 latin square designs

A Latin square design whose automorphism group is transitive of rank at most 3 on points must come from the multiplication table of an elementary abelian p-group, for some prime p.

Homogeneous and Ultrahomogeneous Linear Spaces

A linear spaceSishomogeneousif, whenever the linear structures induced on two finite subsetsS? andS? are isomorphic, there is at least one automorphism ofSmappingS? ontoS?. If every isomorphism fromS? toS? can be extended to an automorphism ofS,Sis calledultrahomogeneous. We give a complete classification of all homogeneous (resp. ultrahomogeneous) linear spaces, without making any finiteness assumption on the number of points ofS.

On geodesic transitive graphs

The main purpose of this paper is to investigate relationships between three graph symmetry properties: s -arc transitivity, s -geodesic transitivity, and s -distance transitivity. A well-known result of Weiss tells us that if a graph of valency at least 3 is s -arc transitive then s ? 7 . We show that for each value of s ? 3 , there are infinitely many s -arc transitive graphs that are t -geodesic transitive for arbitrarily large values of t . For 4 ? s ? 7 , the geodesic transitive graphs that are s -arc transitive can be explicitly described, and all but two of these graphs are related to classical generalized polygons. Finally, we show that the Paley graphs and the Peisert graphs, which are known to be distance transitive, are almost never 2-geodesic transitive, with just three small exceptions.

Involution graphs where the product of two adjacent vertices has order three

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G = PSL(2, q). 2000 Mathematics subject classification: 20B25, 05C25.

Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.

A classification of finite partial linear spaces with a primitive rank 3 automorphism group of grid type

A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is neither a graph nor a linear space is called proper. The aim of this paper is to classify the finite proper partial linear spaces with a primitive rank 3 automorphism group of grid type.