Abstract |
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An algebraic group is a mathematical concept finding its origins in the theory of Lie groups, which nowadays plays a central role in theoretical physics. This mathematical concept is best formulated using the modern language of algebraic geometry, thus making algebraic groups very much algebraic in nature. But when considered over local fields, algebraic groups acquire an interesting topology, and it is then natural to wonder how this topology interacts with the algebraic structure. This line of inquiry has already been explored in the theory of Lie groups, yielding the famous topological characterization of Lie groups by A. Gleason and H. Yamabe in the 50’s. Revolving around this theme, our results are organized in three chapters. The first one aims for a structure of totally disconnected locally compact groups having a linear open subgroup. The second chapter studies Chabauty limits of quasi-split simple algebraic groups acting on trees. Finally, the last chapter gives a condition for the group of semilinear automorphisms of a semisimple group G to decompose as a semidirect product of the group of algebraic automorphisms and the group of field automorphisms preserving G. |