Generic dense free subgroups of the isometry group of the Urysohn space are NSS

Abstract: The isometry group of the bounded Urysohn space, $G = \mathrm{Iso}(\U{1})$ is a central object in the study of Polish groups and topological dynamics. It is known that generic sequences in $G$ generate algebraically free dense subgroups. In this paper, we show that such generic free subgroups exhibit strong geometric rigidity. Specifically, we prove that for a comeager set of sequences generating dense free subgroups $F\leq G$, every non-trivial element $h\in F$ acts with maximal metric displacement, satisfying $\sup_{n\in \N} d(h^n(x),x) = 1$ for every $x \in \U{1}$. As a consequence, these generic subgroups satisfy the \emph{no small subgroup} ($\nss$) property. We note that the method naturally extends to the full isometry group $\mathrm{Iso}(\mathbb{U})$ of the classical Urysohn space.

• The paper establishes that comeagerly many sequences generate free subgroups in the Urysohn space's isometry group that exhibit the NSS property with maximal displacement.
• It employs a blend of descriptive set theory, model theory, and group theory to rigorously control displacement metrics of non-trivial elements.
• The findings imply that generic dense free subgroups are topologically rigid, precluding the presence of small subgroups and contrasting with SSGP behavior.

Generic Dense Free Subgroups of the Isometry Group of the Urysohn Space Are NSS

Introduction and Context

The study investigates the algebraic and topological properties of generic free subgroups within the isometry group $G = \mathrm{Iso}(\mathbb{U}_1)$ of the bounded Urysohn space. The Urysohn space $\mathbb{U}_1$ and its unbounded counterpart $\mathbb{U}$ are central in the theory of Polish groups, serving as universal metric spaces characterized by ultrahomogeneity and universality. Previous results, particularly those by Kechris and Rosendal, have established that generic sequences in $G$ generate algebraically free and dense subgroups, illuminating a rich "generic freeness" phenomenon in these settings.

This work addresses a further topological refinement: the so-called no small subgroup ($NSS$) property, which plays a crucial role in topological group theory and is intimately related to Hilbert’s Fifth Problem. The $NSS$ condition stipulates that a neighborhood of the identity in a topological group contains no non-trivial subgroup, marking a strong form of geometric rigidity. The opposing property, the small subgroup generating property ($SSGP$), has been extensively employed in the context of minimal almost periodicity ($MinAP$) and related group-theoretic constructions.

Main Results

The paper establishes that comeagerly many sequences in $G^N$—i.e., generic choices—generate algebraically free, topologically dense subgroups $F \leq G$ whose non-trivial elements satisfy a strong geometric escape property: for each $\mathbb{U}_1$0 and each point $\mathbb{U}_1$1, the supremum of the displacements $\mathbb{U}_1$2 attains the maximal possible value 1. This immediately implies that such generic free subgroups exhibit the NSS property and, thus, are not SSGP.

Formally, the principal theorem asserts:

For a comeager set of sequences in $\mathbb{U}_1$3, the generated free subgroup $\mathbb{U}_1$4 is NSS and each non-trivial $\mathbb{U}_1$5 displaces every point $\mathbb{U}_1$6 arbitrarily close to the metric diameter.

Strong Numerical Claim

It is explicitly shown that for each non-trivial $\mathbb{U}_1$7,

$\mathbb{U}_1$8

This bound is sharp with respect to the metric diameter.

Moreover, as SSGP is incompatible with NSS, the space of sequences generating SSGP subgroups in $\mathbb{U}_1$9 is meager.

Proof Techniques

The argument integrates methods from descriptive set theory (Baire category), model theory (ultrahomogeneity and universality of the Urysohn space), and group theory (free subgroup structure and displacement metrics).

A key construction is the realization of reduced words as isometries acting with controlled and ultimately maximal metric displacement. For any non-trivial word $\mathbb{U}$0 in the free group on a countable set, and any open subset of the product group, there exist generators yielding isometries such that some power of the image of $\mathbb{U}$1 displaces a basepoint by more than any prescribed $\mathbb{U}$2. This leverages combinatorial and metric amalgamation properties specific to $\mathbb{U}$3.

The Baire category argument, central to establishing genericity, shows that the set of sequences failing this maximal displacement property is meager. Since both the algebraic freeness and the escape property are comeager, their intersection is comeager, providing a robust sense in which the result holds for "most" dense free subgroups.

The extension to the full (unbounded) Urysohn space $\mathbb{U}$4 proceeds mutatis mutandis, with the supremum taken to infinity rather than 1, due to the lack of diameter bounds.

Implications and Theoretical Consequences

This result delineates the landscape of generic subgroup structure within large Polish groups, particularly those evidenced in universal metric spaces. It demonstrates that the free subgroups occupying a comeager portion of the isometry group are not only algebraically generic but also exhibit strong rigidity topologically.

From a topological dynamics vantage, these findings imply that generic actions are maximally non-amenable in the sense of displacement: every non-trivial group element acts without fixed points in any small neighborhood, precluding the existence of non-trivial small subgroups.

The fact that generic dense free subgroups are NSS refutes the possibility that generic such groups might be SSGP, thereby answering an open question in the negative. This further establishes a subtle barrier between algebraic genericity (dense freeness) and topological minimality (SSGP).

Future Perspectives

Given the central place of Polish groups and Urysohn spaces in modern model theory and descriptive set theory, these results illuminate possible directions for the study of automorphism groups in other homogeneous structures. Potential extensions include analogous generic rigidity phenomena for automorphism and isometry groups of generalized metric or relational Fraïssé limits.

A deeper exploration of the boundary between NSS and various forms of local geometric flexibility (such as SSGP and MinAP) within large topological groups could yield further structural classification results. Additionally, these techniques may inform the study of extreme amenability, turbulence, and the fine structure of automorphism groups in logic and dynamics.

Conclusion

The paper provides a rigorous demonstration that generic dense free subgroups of the isometry group of the bounded (and unbounded) Urysohn space are NSS. This establishes that these topologically typical free subgroups exhibit maximal displacement properties for all non-trivial elements, thereby excluding the small subgroup generating property and reinforcing the rigidity of the generic subgroup landscape in universal Polish metric groups. These results clarify the interplay between algebraic freeness, topological density, and geometric rigidity in non-locally compact groups, with significant implications for both abstract group theory and the dynamics of infinite-dimensional symmetric spaces.