Centralisers of fundamental subgroups

A central problem in synthetic geometry is the characterisation of graphs and geometries. The local recognition of locally homogeneous graphs forms one category of such characterisations, which works as follows: Let Δ be a graph. A graph Γ is called locally Δ if for each vertex x of Γ the graph Γx is isomorphic to Δ, where Γx is the induced subgraph of Γ on the set of vertices adjacent to x. It is a natural question to ask for all (connected) graphs, which are locally some graph Δ. This classification question is called the local recognition problem for graphs that are locally Δ, which can be found in great quantities in the literature. One of the earliest and most influential is [1]. The present thesis solves a number of open local recognition problems for different graphs and geometries. First we give an elementary and self-contained charaterization of connected graphs that are locally isomorphic to the line graph of an n-dimensional unitary vector space over the complex numbers endowed with an anisotropic form for n greater or equal to seven. Then we study the graph on the hyperbolic lines (non-degenerate lines) of a unitary vector space over a finite field of dimension at least seven, where two different hyperbolic lines are adjacent if and only if one hyperbolic line is in the polar space of the other hyperbolic line. We prove that the diameter of these graphs is two. This fact enables us to classify all these graphs by their internal properties via the theorem 1.2 of [2]. Finally a local recognation theorem for connected graphs, which are locally isomorphic to the line graph of the six-dimensional unitary vector space over the complex numbers endowed with an anisotropic form, is proved. We describe in the last part a method to construct some subgroup of the automorphism group of the considered graphs with certain properties for each vertex of the graph and its neighborhood. Moreover a classification of this subgroup of automorphisms is possible via a locally reflection graph and the theory of weak Phan systems. This classification of the constructed subgroup implies a local recognation of the universal cover of the graphs. [1] Francis Buekenhout, Xavier Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra 45 (1977), 391--434 [2] Hans Cuypers, The geometry of hyperbolic lines in polar spaces, unpublishe