An epimorphic subgroup arising from Roberts’ counterexample

In 1994, based on Roberts’ counterexample to Hilbert’s fourteenth problem, A’Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H0H0 on a 19-dimensional vector space VV such that the algebra of invariants k[V]H0k[V]H0 is not finitely generated. We consider a certain extension HH of H0H0 by a one-dimensional torus and prove that HH is epimorphic in SL(V). In particular, the homogeneous space SL(V)/H provides a new example of a homogeneous space with epimorphic stabilizer that admits no projective embeddings with small boundary.