Alternating and Symmetric Separability of Free Products

Abstract: Let $F \ast G$ be a free product of a free group $F$ and a LERF group $G$. In this note, we provide sufficient conditions for a subgroup $H$ of $F \ast G$ to be $\mathcal{A} \cup \mathcal{S}$-separable, that is, for any finite set ${γ_1, \ldots, γ_n} \subset (F \ast G) \setminus H$, there is a surjection $f$ from $F \ast G$ to an alternating or symmetric group such that $f(γ_i) \notin f(H)$ for all $i$. As a corollary, any finitely generated infinite-index subgroup of a free group is $\mathcal{A} \cup \mathcal{S}$-separable in the free product of the free group and an arbitrary LERF group, generalizing a result of Wilton.

• The paper demonstrates that finitely generated subgroups in free products exhibit alternating and symmetric separability under specific trivial intersection or infinite-index conditions.
• It employs labeled graphs, covering space theory, and permutation group methods to construct explicit separating homomorphisms onto alternating and symmetric groups.
• The work generalizes Wilton’s result from free groups to free products, offering significant insights into subgroup separability in geometric group theory.

Alternating and Symmetric Separability in Free Products of LERF Groups

Overview

The paper "Alternating and Symmetric Separability of Free Products" (2604.17232) systematically investigates conditions under which finitely generated subgroups of free products involving a free group $F$ and a LERF group $G$ are alternating or symmetric separable. The study leverages topological and combinatorial techniques, particularly labeled graphs (based on the Scott–Markus-Epstein framework) and covering space theory, to provide criteria for $\cup$-separability, where quotients are realized within the alternating or symmetric groups. Importantly, this extends Wilton’s result on alternating separability of infinite-index free group subgroups to broader free product settings.

Context: Residual Properties and Separability

LERF (locally extended residually finite) groups form a pivotal class in geometric group theory due to their connection with subgroup separability. A group $G$ is LERF if for any finitely generated $H < G$ and $\gamma \in G \setminus H$, there exists a finite quotient in which $\gamma$ is not in the image of $H$. Free groups, surface groups, and many 3-manifold groups are LERF; this property fails for certain higher-dimensional manifolds and graph manifold groups.

Residual properties, such as residual finiteness and separability, are often refined by restricting the finite quotients to specific simple groups, notably the alternating groups $A_n$ and symmetric groups $S_n$. This yields subgroup properties such as alternating-separability and symmetric-separability, requiring discriminating maps onto $G$0 or $G$1 for subgroup avoidance.

Wilton [Wi, 2012] established that any finitely generated, infinite-index subgroup of a free group is alternating-separable, a result later extended to surface groups and RA Coxeter groups [Bu, 2021]. The present paper seeks analogous separability in free products $G$2 with $G$3 LERF.

Main Theorem and Structural Criteria

The central theorem establishes the following:

Given $G$4 free of rank $G$5 and $G$6 LERF, a finitely generated subgroup $G$7 is $G$8-separable provided at least one of:

1. Trivial intersection: $G$9 for all $\cup$0.
2. Infinite index intersection: There exists $\cup$1 with $\cup$2 and $\cup$3.

This statement strictly generalizes Wilton’s result from $\cup$4 to $\cup$5 and holds for all finitely generated infinite-index subgroups of $\cup$6 in $\cup$7 with LERF $\cup$8.

A critical corollary is that every finitely generated, infinite-index subgroup of $\cup$9 embeds with $G$0-separability in $G$1, for arbitrary LERF $G$2.

Technical Framework

Labeled Graphs and Precovers

The approach utilizes Stallings-type labeled graphs, relative Cayley graphs, and precovers (see [Ma2]), enabling an explicit description of subgroup embeddings and separability certificates. Subgroups correspond to labels of based loops in a graph, and their separability hinges on embedding finite graphs into covers associated with finite quotients.

The embedding machinery ensures that, for given subgroup data and elements to separate, a finite labeled graph (encoding coset data and subgroup generators) can be embedded in a finite cover; further, this cover’s automorphism group can be forced to be $G$3 or $G$4 by constructions inspired by Wilton.

Jordan’s Theorem and Alternating/Symmetric Action

The proof exploits Jordan’s classical theorem on primitive permutation groups: If a large enough prime degree permutation group is transitive and has an element with bounded support, it must be $G$5 or $G$6. This enables the construction of separating homomorphisms whose image is guaranteed to be alternating or symmetric.

Key steps involve:

• Creating a labeled graph corresponding to the subgroup and words to separate.
• Pushing out the relevant monochromatic components into actual covers (using LERFness of factors).
• Modifying a non-cover $G$7-component via Wilton’s gadgets (e.g. $G$8, $G$9 graphs) to guarantee that the covering action on vertices is alternating or symmetric.
• Assembling these into a precover of $H < G$0, then embedding in an honest finite cover (which corresponds, via the action on cosets, to a separating surjection onto $H < G$1 or $H < G$2).

By showing that the induced permutation action meets Jordan’s primitivity and bounded support criterion, the authors ensure the quotient is $H < G$3 or $H < G$4, and unwanted elements avoid the image of $H < G$5.

Kurosh Theorem and Decomposition

The paper combines the graphical precover construction with the Kurosh subgroup theorem, which describes arbitrary subgroups of free products as free products of conjugates of free factor subgroups and a free factor. The Markus-Epstein algorithm enables reading the Kurosh decomposition directly from the graph, identifying the required monochromatic (factor) components.

Numerical and Structural Results

While the results are primarily qualitative and structural, the proof method is constructive: for each tuple to separate, a finite cover with degree a large prime $H < G$6 is produced, and the support of a specified generator on this cover is explicitly bounded (by the size of the constructed labeled graph and auxiliary components). This aligns with Wilton’s bounds and ensures uniform effectiveness in the construction of separating quotients.

The paper also presents an explicit counterexample demonstrating that finite-index subgroups need not be alternating separable in the free product setting, highlighting the necessity of the infinite-index hypothesis in $H < G$7.

Theoretical and Practical Implications

The paper’s results provide a general framework for analyzing alternating/symmetric separability in mixed free products, with direct implications for subgroup separability phenomena in amalgam and HNN extension contexts. The approach, rooted in the interplay between group actions, topological covers, and finite simple group quotients, has potential applicability to low-dimensional topology (e.g., covering space theory, 3-manifold group properties) and algorithmic group theory (algorithmically constructing quotients witnessing separability).

Future directions include tightening the structural understanding of separating quotients for broader classes (e.g., limit groups, virtual special groups), characterizing finite quotients with controlled simple structure, and leveraging the presented framework in the study of profinite rigidity and decision problems related to separability.

Conclusion

The paper provides a comprehensive analysis of alternating and symmetric separability properties for finitely generated subgroups in free products $H < G$8 with LERF factors. Via labeled graph techniques, covering space constructions, and permutation group theory, it extends known results from free groups to this broader class, with structural criteria that are both necessary and sufficient in natural settings. The interplay of graph topology and finite group quotients in the separability proofs is elegant and effective, and the results significantly generalize previous findings regarding subgroup separability in free (and related) groups.