On the automorphism group of the asymptotic pants complex of a planar surface of infinite type

We consider a planar surface \Sigma of infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of \Sigma and prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z/2Z.

💡 Research Summary

The paper investigates a planar surface Σ of infinite type whose asymptotic mapping class group coincides with the Thompson group T. The authors introduce a new combinatorial object, the asymptotic pants complex C, built from pants decompositions of Σ that are “asymptotically” defined: each decomposition consists of infinitely many simple closed curves whose ends accumulate on the Cantor set of ends of Σ. Vertices of C correspond to such decompositions, while edges represent elementary “flips” that replace a single curve by another curve intersecting it minimally. Two flips that commute give rise to a square 2‑cell, so C is a 2‑dimensional flag complex with a locally infinite but highly regular structure.

The first major result is that the Thompson group T acts on C by simplicial automorphisms and that this action is transitive on vertices. The proof exploits the well‑known tree‑pair diagram description of T. Each element of T induces a homeomorphism of the end space of Σ, which in turn permutes the curves of any pants decomposition in a controlled way. By constructing a finite sequence of flips that connects any two given decompositions and showing that this sequence can be realized by a suitable element of T (using a “delayed flip” technique to handle infinitely many curves simultaneously), the authors establish vertex‑transitivity. Moreover, the action respects the square relations, so it extends to a full simplicial action on C.

The second main theorem determines the full automorphism group Aut(C). The authors first prove a rigidity statement: any simplicial automorphism of C must preserve the flip structure and therefore arises from a homeomorphism of Σ that respects the asymptotic end structure. Since the asymptotic mapping class group is exactly T, every orientation‑preserving automorphism of C lies in the image of T. The only additional automorphisms are orientation‑reversing involutions that globally flip the surface, which form a group isomorphic to ℤ/2ℤ. By analyzing how this involution interacts with T (it does not commute with all elements), they show that Aut(C) fits into a short exact sequence

 1 → T → Aut(C) → ℤ/2ℤ → 1

which is non‑split; equivalently, Aut(C) is a semidirect product of T by ℤ/2ℤ. This description mirrors similar results for curve complexes of finite‑type surfaces, but the infinite‑type setting requires new combinatorial arguments to control the infinite families of curves.

The paper concludes with a discussion of implications. The asymptotic pants complex provides a natural geometric model for the asymptotic mapping class group of Σ, analogous to the classical pants complex for finite‑type surfaces. Its automorphism group being an extension of T by a single involution highlights a striking rigidity: despite the infinite complexity of Σ, the only symmetries of the associated combinatorial structure are those already encoded in the algebraic structure of T, together with a global orientation reversal. The authors suggest several directions for future work, including extending the construction to other infinite‑type surfaces (e.g., surfaces with infinitely many punctures but different end space topologies), investigating higher‑dimensional analogues (such as asymptotic flip complexes for higher‑genus surfaces), and exploring connections with the representation theory of T and with homological stability phenomena in infinite‑type mapping class groups. Overall, the work bridges the gap between the algebraic world of Thompson’s groups and the geometric topology of infinite‑type surfaces, offering a new tool for understanding their symmetries.