On compression of Bruhat-Tits buildings

We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type \(A\). More precisely, consider a \(p\)-adic linear space \(V\) and the set \(Lat(V)\) of all lattices in \(V\). The complex distance in \(Lat(V)\) is a complete system of invariants of a pair of points of \(Lat(V)\) under the action of the complete linear group. An element of a Nazarov semigroup is a lattice in the duplicated linear space \(V\oplus V\). We investigate behavior of the complex distance under the action of the Nazarov semigroup on the set \(Lat(V)\).