Property $P_{ ext{naive}}$ for big mapping class groups

We study the property $P_{\text {naive }}$ of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements $h_{1},h_{2}, \cdots, h_{n}$, there exists another element $g\neq 1$ of infinite order such that for all $i$, $\langle g, h_{i}\rangle \cong \langle g \rangle * \langle h_{i} \rangle$.

💡 Research Summary

The paper investigates the “P‑naive” property for mapping class groups of infinite‑type surfaces. A group G has property P‑naive if for every finite set of non‑trivial elements F⊂G there exists an infinite‑order element g∈G such that for each h∈F the subgroup ⟨g,h⟩ is isomorphic to the free product ⟨g⟩∗⟨h⟩. This property, introduced by Bekk‑a, Cowling and de la Harpe, is known to imply simplicity of the reduced C∗‑algebra C∗_r(G). It holds for many acylindrically hyperbolic groups because they contain many WWPD loxodromic elements, which allow a ping‑pong argument.

Infinite‑type surfaces S are never acylindrically hyperbolic, so the usual machinery does not apply. The author therefore exploits the notion of a nondisplaceable subsurface, originally defined by Mann and Rafi. A connected finite‑type subsurface K⊂S is nondisplaceable if for every homeomorphism f∈Homeo(S) we have f(K)∩K≠∅. The existence of such a K imposes strong topological constraints on S and provides a foothold for hyperbolic dynamics.

The main theorem (Theorem 1.1) states: if S is a connected, orientable surface of positive complexity that contains a nondisplaceable connected finite‑type subsurface, then its mapping class group Map(S) satisfies property P‑naive. The proof proceeds in several stages.

1. Construction of a suitable nondisplaceable subsurface. Lemmas 2.3–2.5 show that given any finite collection of mapping classes {h_i}, one can enlarge a given nondisplaceable subsurface K′ to a larger nondisplaceable K that contains all essential curves moved by the h_i and such that none of the h_i restricts to the identity on K. This ensures that each h_i acts non‑trivially on the curve graph of K.

2. Selection of a K‑pseudo‑Anosov element. Inside the finite‑type subsurface K, pick a pseudo‑Anosov mapping class and extend it by the identity outside K. The resulting element g is called K‑pseudo‑Anosov. By a result of Horbez‑Qing‑Rafi, such an element is WWPD loxodromic for the hyperbolic space constructed from the projection complex associated to the orbit of K.

3. Building a hyperbolic space via projection complexes. For each translate K_i of K under Map(S) one considers the curve graph C_S(K_i) consisting of essential curves lying in K_i. Using the Bestvina‑Bromberg‑Fujiwara (BBF) projection machinery, one defines projection maps π_{K_i}(K_j)⊂C_S(K_i) and verifies the three projection axioms (P1)–(P3). The resulting projection complex C(Y_K) is a quasi‑tree (Theorem 2.7) and, because each C_S(K_i) is hyperbolic, C(Y_K) is a δ‑hyperbolic space on which Map(S) acts continuously, isometrically, and nonelementarily.

4. Ping‑pong argument to obtain free products. The element g is a WWPD loxodromic isometry of C(Y_K); its axis provides two disjoint “ping‑pong” domains. Each h_i, by construction, does not fix any point of K and therefore acts on C(Y_K) without preserving the axis of g. Applying the Dahmani‑Guirardel‑Osin ping‑pong lemma for groups acting on hyperbolic spaces, one deduces that for each i the subgroup ⟨g,h_i⟩ is a free product ⟨g⟩∗⟨h_i⟩. Since the choice of g depends only on the finite set {h_i}, the property P‑naive follows.

The paper also discusses the broader context: property P‑naive is a strong form of dynamical flexibility, and its presence in Map(S) suggests that even though infinite‑type mapping class groups are not acylindrically hyperbolic, they retain enough hyperbolic behavior when a nondisplaceable finite‑type subsurface exists. Consequently, the reduced C∗‑algebra of such a mapping class group is expected to be simple, extending known results for acylindrically hyperbolic groups.

In summary, the author proves that the existence of a nondisplaceable finite‑type subsurface in an infinite‑type surface guarantees that its mapping class group possesses the P‑naive property. The proof blends topological arguments about subsurfaces, the BBF projection complex machinery, and hyperbolic group theory (WWPD elements and ping‑pong). This work opens the door to applying C∗‑simplicity and related rigidity phenomena to a new class of “big” mapping class groups that were previously inaccessible via standard acylindrical hyperbolicity techniques.