Strongly bounded generation in transformation groups
Nicholas G. Vlamis
TL;DR
This work develops an algebraic counterpart to Rosendal's CB-generation by defining $SB$-generated groups and proving that several natural transformation groups are $SB$-generated. It shows that $SB$-generated groups admit maximal left-invariant metrics and have uncountable cofinality, and establishes $SB$-generation for $\mathrm{Homeo}_0(M)$ (closed $M$), certain big mapping class groups, and $\mathrm{Homeo}(X)$ for compact well-ordered spaces with successor limit capacity. The results extend finite generation while preserving large-scale rigidity, yielding consequences such as finitely generated abelianizations and connections to unique Polish topologies in specific CB-generated Polish groups. Overall, the paper links large-scale geometric invariants, algebraic generation, and the topology of transformation groups across manifolds, infinite-type surfaces, and ordinal-end spaces, providing a robust framework for understanding strongly bounded generation in natural contexts.
Abstract
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.