Journal of Symplectic Geometry

Volume 21 (2023)

Number 2

Arboreal models and their stability

Pages: 331 – 381



Daniel Álvarez-Gavela (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.)

Yakov Eliashberg (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

David Nadler (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.)


The main result of this paper is the uniqueness of local arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. A parametric version implies that the space of germs of symplectomorphisms that preserve the local model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically. This paper can be read independently, but it is part of a series of papers by the authors on the arborealization program.

D.A. was partially supported by NSF grant DMS-1638352 and the Simons Foundation.

Y.E. was partially supported by NSF grant DMS-1807270.

D.N. was partially supported by NSF grant DMS-1802373.

Received 19 November 2021

Received revised 28 September 2022

Accepted 2 October 2022

Published 28 September 2023