The Neretin groups

This is the extended version of the talk we gave during the Arbeitsgemeinschaft on totally disconnected locally compact groups, held in Oberwolfach in October 2014. We give the definition of the Neretin group of spheromorphisms of the infinite $q$-regular tree, and describe its natural totally disconnected group topology, which is not the one that usually appears as the first guess. We also present the proof of simplicity due to Kapoudjian. We do not make any claims concerning the originality of the presented results.