A lower bound for the canonical dimension of compactly induced representations

The canonical dimension is an invariant attached to admissible representations of p-adic reductive groups, which has only received significant attention in the case of mod-p representations. In the case of complex representations, the canonical dimension is closely related to the wavefront set. We find a new lower bound for the canonical dimension of a general compactly induced representation over an arbitrary coefficient field. This lower bound is uniform in the sense that it only depends on the group and not on the representation itself. In many cases, this provides a lower bound for the canonical dimension of supercuspidal representations and in the complex case we get a lower bound for the corresponding wavefront set. In order to obtain this result, we first generalize a result on the asymptotic growth of the cardinality of balls in the Bruhat-Tits building to the case of exceptional types.