To each totally disconnected, locally compact topological group G and eachgroup A of automorphisms of G, a pseudo-metric space of ``directions'' has beenassociated by U. Baumgartner and the second author. Given a Lie group G over alocal field, it is a natural idea to try to define a map from the space ofdirections of analytic automorphisms of G to the space of directions ofautomorphisms of the Lie algebra L(G) of G, which takes the direction of ananalytic automorphism of G to the direction of the associated Lie algebraautomorphism. We show that, in general, this map is not well-defined. However,the pathology cannot occur for a large class of linear algebraic groups (called``generalized Cayley groups'' here). For such groups, the assignment justproposed defines a well-defined isometric embedding from the space ofdirections of inner automorphisms of G to the space of directions ofautomorphisms of L(G). Some counterexamples concerning the existence of smalljoint tidy subgroups for flat groups of automorphisms are also provided.
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