math.GR — Group Theory

Finite-length permutation modules for\\free-amalgamation structures of arbitrary arity

Contributed by Dr. András Pongrácz

Let $\A$ be the countable homogeneous limit of a free-amalgamation class in a finite relational language, expanded by a generic linear order. We prove that its tuple permutation modules have finite composition length over every field, also after naming finitely many points. More precisely, each orbit of increasing $d$-tuples disjoint from the named points has a permutation module of length $2^d$. This extends the theorem of Yang--Boja\'nczyk--Klin from unary and binary relations to relations of arbitrary arity. The main step shows that every finitely supported array whose sums over each single coordinate vanish is a linear combination of alternating sums obtained by independently doubling its coordinates. Its proof combines a Ramsey separation theorem with a construction that moves finite configurations into prescribed order intervals. Over a finite field, we also obtain finitely many closed invariant linear subspaces of $\F^{A^k}$; each is generated, in the product topology, by functions definable using at most $k$ additional parameters.