Towards a Stallings-type theorem for finite groups

TL;DR

The paper develops a local approach to Stallings-type decompositions for finite, nilpotent groups by analyzing r-local separators and r-local coverings of Cayley graphs. The authors introduce a robust local framework—local cutvertices, local 2-separations, traversals, and morphemes—to translate local separation data into global group structure, overcoming the lack of ends in finite groups. The central result shows that for a finite nilpotent group $ig\Gammaigrn$ of class $ig $ and sufficiently large radius $r$, an $r$-local separator of size at most $2$ implies either $ig angle oldsymbol{ eq} angle$ a cyclic group of order $>r$ or $ig angle oldsymbol{ eq} angle$ a direct product $ig(C_i imes C_jig)$ with $i>r$ and $j\\in\{1,2\}$, paralleling Stallings’ ends decomposition in a local setting. The proof splits into non-involution and involution cases, using an Edge Insertion Lemma to augment generating sets, a tower of local 2-separation lemmas, and intricate traversals of iterated commutators to force the claimed product structures. The work lays groundwork toward extending Stallings-type results to broader finite groups via local locality concepts, while noting obstructions for higher-order local separations and non-nilpotent cases. Overall, the paper connects local combinatorial structure to global algebraic decomposition in finite nilpotent groups through a novel use of local coverings and Morpheme/Magic techniques.

Abstract

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local covering: a usually infinite graph obtained by keeping all local structure of the original graph while unfolding all other structure as much as possible. We use local separators and local coverings to discover and prove a low-order Stallings-type result for finite nilpotent groups $Γ$: the $r$-local covering of some Cayley graph $G$ of $Γ$ has $\geq 2$ ends that are separated by $\leq 2$ vertices iff $G$ has an $r$-local separator of size $\leq 2$ and $Γ$ has order $>r$, iff $Γ$ is isomorphic to $C_i\times C_j$ for some $i>r$ and $j\in\{1,2\}$.

Figures (18)

• Figure 1: Bottom: The Cayley graph $G$ of $C_{12}\times C_2=:\Gamma$. The twelve circled vertex-sets are the separators of a set $N$ of pairwise nested 11-local 2-separations of $G$. The group $\Gamma$ acts transitively on $N$. Every element of $N$ lifts to a 2-separation of the 11-local covering of $G$ (top) where it separates two ends.
• Figure 2: Every vertex of the Cayley graph of $A_5\cong\langle a,b:a^3,b^2,(ab)^5\rangle$ is a 9-local cutvertex. However, these local cutvertices cannot give rise to interesting product structure, as alternating groups $A_n$ are simple for $n\geqslant 5$. More generally, for every $r>0$ there is $n$ such that every vertex of some Cayley graph of $A_n$ is an $r$-local cutvertex, see \ref{['UnboundedLocalCutvertex']}.
• Figure 3: In the proof of \ref{['lem:short_cycle_forall']} we construct a walk by concatenating three orange paths joining four neighbours of $\mathbb{I}$.
• Figure 4: An $a$--$b$ walk that traverses a local $2$-separator $X$ oddly. The traversals are bold.
• Figure 5: Two crossing $r$-local separations as in \ref{['empty_corner']}, where $v$ is not an $r$-local cutvertex

Theorems & Definitions (139)

• Theorem 1
• Definition 1: Morpheme
• Example 1
• Lemma 1: Folklore
• proof
• Lemma 2: Morpheme Finding Lemma
• proof
• Definition 2: Iterated commutator word
• Example 2
• Definition 3: Truncation