  We study the scale function, space of directions and scale-multiplicative
semigroups for restricted Burger-Mozes groups. We relate these general notions
to intrinsic properties of the group. Among other things, we give a formula for
the scale function; relate the space of directions to both the action on the
tree and an action on a $\operatorname{CAT}(0)$ cube complex; and construct
maximal scale-multiplicative semigroups from the space of directions.

Group Theory