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An excursion into the $p$-adic world. (Ein Ausflug in die $p$-adische Welt.) (German)
Wendland, Katrin (ed.) et al., Facettenreiche Mathematik. Einblicke in die moderne mathematische Forschung für alle, die mehr von Mathematik verstehen wollen. Wiesbaden: Vieweg+Teubner (ISBN 978-3-8348-1414-2/pbk; 978-3-8348-8173-1). Mathematik Populär, 433-451 (2011).
This article provides readers with a brief introduction to topics covered in {\it J.-P. Serre}’s book [Trees. Springer Monographs in Mathematics. Berlin: Springer. (2003; Zbl 1013.20001)]. It begins by explaining $p$-adic absolute values, $p$-adic lattices $L$ in the plane, the action of $\mathrm{GL}_2(\Bbb Q)$ on these lattices, and equivalence classes of lattices with respect homothety. By defining when two lattices are “neighbors” one is led to the definition of a graph whose vertices are (equivalence classes of) lattices, which are connected by an edge if they are neighbors of each other. The main result is that these graphs $X(p)$ are trees, and that it is possible to study the group $\mathrm{SL}_2(\Bbb Q)$ by investigating its action on this tree.
Reviewer: Franz Lemmermeyer (Jagstzell)
Classification: F60