Is there an infinite finitely presented simple amenable group?

Theorem 1.1: there exists an infinite finitely presented simple amenable group. It is an alternating subgroup of a polygon exchange group (piecewise translations of finitely many squares over a real quadratic ring, with two inclined edge directions besides the coordinate ones). Finite presentation comes from propagating relations over a fixed finite generating set with a homological argument in the style of Szymik-Wahl; amenability from correlated random polygonal partitions giving almost-invariant measures. No explicit bound on the sizes of the generating set or relators is given.

Theorem 1.1: there exists an infinite finitely presented simple amenable group. It is an alternating subgroup of a polygon exchange group (piecewise translations of finitely many squares over a real quadratic ring, with two inclined edge directions besides the coordinate ones). Finite presentation comes from propagating relations over a fixed finite generating set with a homological argument in the style of Szymik-Wahl; amenability from correlated random polygonal partitions giving almost-invariant measures. No explicit bound on the sizes of the generating set or relators is given.