The Smooth Void

An irreducible representation is indivisible: you cannot split it into simpler pieces. Smooth vectors — those that behave well under the group action — are usually where the algebraic structure lives. In well-behaved settings, the smooth vectors of an irreducible representation contain at least one algebraically irreducible component. The global indivisibility has a local witness.

Monod (arXiv:2603.16819) shows this fails for automorphism groups of regular locally finite trees. These groups admit irreducible Banach space representations where the smooth vectors contain no algebraically irreducible component at all. The representation is indivisible as a whole, but every attempt to extract a clean algebraic piece from its smooth interior comes up empty.

This is a strange kind of structure. The representation is a single, unified object — you cannot decompose it. But zoom into the smooth part and you find no irreducible core, no algebraic atom. The unity exists only at the topological level. At the algebraic level, the representation is everywhere divisible, infinitely complex, yet the whole is simple.

The tree automorphism groups are the key. Trees have a rigid combinatorial structure — every path between two vertices is unique — but their automorphism groups are wild: uncountable, non-discrete, rich with symmetry. The infinite branching of the tree propagates into infinite algebraic complexity in the smooth vectors, while the global topology of the representation remains simple.

The result means you cannot always recover algebraic structure from topological structure, even in the most natural way (passing to smooth vectors of an irreducible representation). The smooth vectors are supposed to be where the algebra lives. Here, the algebra has vacated the premises while the topology stands firm. Unity at one scale masks infinite complexity at another.