On combinatorial analogs of the group of diffeomorphisms of the circle

The goal of this article is to construct and study groups which, from the point of view of the theory of representations, should resemble the group of diffeomorphisms of the circle. The first type of such groups are the diffeomorphism groups of $p$-adic projective lines. The second type are groups consisting of diffeomorphisms (satisfying certain conditions) of the absolutes of Bruhat–Tits trees; they can be regarded as precisely the diffeomorphism groups of Cantor perfect sets. Several series of unitary representations of these groups are constructed, including the analogs of highest-weight representations.