Strongly bounded generation in transformation groups

Word metrics on finitely generated groups have canonical quasi-isometry classes, making quasi-isometry invariants genuine group invariants. Rosendal generalized this phenomenon to topological groups through CB-generation, but in the general topological setting the resulting quasi-isometry invariants are not invariants of the underlying abstract group. Specializing to the discrete case yields what we call SB-generated groups, where the invariants are genuinely algebraic. We show that SB-generation arises naturally in transformation groups by identifying several broad families of examples: the identity component of homeomorphism groups of closed manifolds, certain big mapping class groups, and homeomorphism groups of compact well-ordered spaces with successor limit capacity. These results demonstrate that SB-generation provides a robust extension of finite generation.

💡 Research Summary

The paper introduces the notion of strongly bounded generation (SB‑generation) for discrete groups and demonstrates that this concept provides a robust algebraic analogue of finite generation, especially in the context of transformation groups. A subset of a group is called strongly bounded if it has finite diameter with respect to every left‑invariant metric on the group. A group is SB‑generated when it can be generated by such a subset. This definition strictly contains all finitely generated groups but also includes many natural infinite groups that are not finitely generated.

The authors first establish the basic metric properties of SB‑generated groups. They prove that any word metric arising from a strongly bounded generating set is maximal: it dominates every other left‑invariant pseudo‑metric up to a linear distortion. Consequently, any two word metrics coming from different strongly bounded generating sets are bi‑Lipschitz equivalent, and the quasi‑isometry class of the metric becomes a genuine invariant of the abstract group. Moreover, a group admits a maximal left‑invariant metric if and only if it is SB‑generated, establishing a precise equivalence (Proposition 2.6).

Structural consequences are explored: an abelian group is SB‑generated precisely when it is finitely generated, and the abelianisation of any SB‑generated group is therefore finitely generated. The class is closed under taking quotients, but not under arbitrary extensions; for example, free products of strongly bounded groups yield SB‑generated groups that are not themselves strongly bounded.

The core of the paper consists of three families of transformation groups where SB‑generation is proved:

1. Identity components of homeomorphism groups of closed manifolds.
   Building on Mann–Rosendal’s result that Homeo₀(M) is CB‑generated (coarsely bounded generated) for a closed manifold M, the author shows that the generating set they construct is actually strongly bounded. Hence Homeo₀(M) is SB‑generated (Theorem 4.5). This strengthens the earlier result and applies even when the group is not strongly bounded (e.g., when dim M ≥ 2 and π₁(M) contains an element of infinite order). As a corollary, if the mapping class group MCG(M) is finitely generated (true for all closed surfaces), then the full homeomorphism group Homeo(M) is SB‑generated.

2. Big mapping class groups of infinite‑type surfaces.
   The paper studies surfaces that are infinite‑type (non‑compact with infinitely many ends). Using the notion of telescoping surfaces and the classification of maximal ends, the author proves a local strong boundedness result (Theorem 5.2): under a symmetry condition on the collection of telescoping pieces, there exists a finite‑type subsurface Σ such that the subgroup U_Σ (mapping classes acting trivially on Σ) is strongly bounded. When the whole mapping class group is CB‑generated (the “tame” surfaces classified by Mann–Rafi), this local property lifts to global SB‑generation (Corollary 1.3). An explicit example is the mapping class group of the plane minus a Cantor set, which becomes an infinite‑diameter Gromov‑hyperbolic SB‑generated group.

3. Homeomorphism groups of compact well‑ordered spaces with successor limit capacity.
   Compact well‑ordered spaces are classified by a limit capacity (an ordinal) and a degree (a natural number). If the limit capacity is a successor ordinal, the author shows that the homeomorphism group is SB‑generated (Theorem 6.1). This yields, for any cardinal κ, a family of pairwise non‑isomorphic SB‑generated groups of size κ that are neither finitely generated nor strongly bounded (Corollary 6.2).

The paper also discusses the relationship between SB‑generation and unique Polish topologies. While many CB‑generated Polish groups (e.g., Homeo₀(M) for manifolds) have a unique Polish group topology, the author provides a counterexample: a CB‑generated Polish group with a unique Polish topology that fails to be SB‑generated because it surjects onto ℚ, violating the finite‑generation of its abelianisation (Proposition 1.5).

Overall, the work demonstrates that SB‑generation furnishes a canonical metric structure on a wide array of transformation groups, turning quasi‑isometric invariants into true algebraic invariants. It extends the geometric group theory toolkit beyond finitely generated groups, opens new avenues for studying large transformation groups, and suggests further investigations into classification, rigidity, and the interplay between algebraic and topological properties of SB‑generated groups.