Theorem Establishes Continuum Many Local Isomorphism Types

Totally disconnected locally compact, or tdlc, groups provide a setting for studying symmetries of locally finite structures. Their local structure is unusually consequential: a compact open subgroup supplies a neighbourhood basis at the identity, and two tdlc groups are locally isomorphic when they have isomorphic compact open subgroups. For the non-discrete, compactly generated, topologically simple members of the class $\mathscr{S}$, the authors address a longstanding counting question: can there be uncountably many local isomorphism types?

The paper's answer is yes. It states that $\mathscr{S}$ contains $2^{\aleph_0}$ pairwise non-locally-isomorphic groups. This is stronger than merely producing many non-isomorphic examples: it separates the groups at the level of compact open subgroups, the invariant that governs local properties in this setting.

Core Contribution

The central theorem asserts the existence of $2^{\aleph_0}$ non-discrete, compactly generated, topologically simple tdlc groups that are pairwise non-locally-isomorphic. The result resolves the open counting question posed for the class $\mathscr{S}$.

The distinction from prior construction techniques is the invariant being controlled. Smith's box-product construction was already known to turn suitable permutation groups into simple tdlc groups, but it does not by itself distinguish local isomorphism classes. Here, the authors begin with an uncountable family of profinite groups whose commensurability relations are rigid enough to survive the subsequent construction. A closed subgroup isomorphic to $G(\beta)$ inside the compact open subgroup $W(\alpha)$ forces $\alpha=\beta$; that implication supplies the local separation.

Technical Approach

The proof outline starts from an uncountable family $\{G(\alpha)\}$ of pairwise non-commensurable uniform pro-$p$ groups, attributed to work of Ilir Snopce. For each parameter $\alpha$ in an uncountable index set $A$, the authors construct an infinite iterated wreath product $W(\alpha)$ involving $G(\alpha)$. The displayed role of this stage is to create an uncountable family of profinite groups that remain pairwise non-commensurable.

They then apply the box-product construction and techniques from Smith's 2017 work to obtain a closed primitive permutation group $S(\alpha)$. The paper states that each $S(\alpha)$ is non-discrete, compactly generated, totally disconnected, and topologically simple, while admitting $W(\alpha)$ as an open subgroup. Thus $W(\alpha)$ is not ancillary data: it is a compact-open witness to the local structure of $S(\alpha)$.

The final step is a rigidity lemma. If $W(\alpha)$ has a closed subgroup isomorphic to $G(\beta)$, then the indices agree. Since local isomorphism would identify sufficiently small compact open subgroups, this property rules out local isomorphism between $S(\alpha)$ and $S(\beta)$ for distinct parameters. The construction packages a profinite commensurability obstruction into a simple tdlc group.

Results and Analysis

The stated output has cardinality $2^{\aleph_0}$, establishing uncountably many local isomorphism classes rather than only infinitely many examples. The paper also places the result against a 2011 structural theorem of Caprace and Monod: non-discrete, compactly generated, topologically simple tdlc groups form the class at the centre of contemporary structural questions about tdlc groups.

For specialists, the value is not a performance comparison but the proof architecture. Pairwise non-commensurability is a stronger starting condition than pairwise non-isomorphism, and the rigidity statement is exactly what is needed to translate that distinction into local non-isomorphism after the box-product step. If the announced full proof establishes this lemma at the stated generality, the result closes the counting question cleanly.

Scope of the Evidence

This four-page preprint is explicitly an overview of a forthcoming paper, not the complete proof. It provides the theorem, construction route, and the required rigidity implication, but does not state the detailed definition of the iterated wreath products, prove the rigidity lemma, or supply the intermediate arguments showing that the resulting permutation groups have every claimed property. The mathematical claim is therefore substantial, but readers seeking a proof must wait for the promised full version.