Polymatroids on Stallings Core Graphs

• The paper introduces a polymatroid framework on Stallings core graphs to encode structural, algebraic, and probabilistic invariants of free group subgroups.
• It unifies classical bounds such as the strengthened Hanna Neumann inequality and Wise’s cyclic bound by associating submodular rank functions to graph data.
• The methods yield new lower bounds on stable invariants and probabilistic upper limits on subgroup mappings via random homomorphisms, bridging group theory and combinatorial optimization.

Polymatroids on Stallings core graphs provide a functional and combinatorial framework for encoding and analyzing structural, algebraic, and probabilistic invariants of subgroups of free groups. This theory reformulates and unifies a variety of classical and recent group-theoretic bounds, especially those related to the Hanna Neumann Conjecture and its generalizations, by associating submodular rank functions—polymatroids—to graph-theoretic, homological, and action-theoretic data on Stallings core graphs. The resulting methods yield new lower bounds on stable invariants for subgroups, and new upper bounds on the probability that words or subgroups map to prescribed subgroups under random homomorphisms into finite groups, bridging group theory, random mapping, invariant theory, and combinatorial optimization (Shomroni, 31 Dec 2025).

1. Stallings Core Graphs and Their Structure

Let $F = \mathrm{Free}(B)$ denote the free group on a finite generating set $B$. For every finitely generated subgroup $H \leq F$, the associated Stallings core graph $\Gamma_H$ is constructed as a finite, $B$-labeled covering of the bouquet $\Omega_B$—the wedge of $|B|$ loops labeled by $B$—with all “hanging trees” and isolated tree components pruned away. The immersion

$\iota: \Gamma_H \longrightarrow \Omega_B$

is uniquely specified by the subgroup $H$. Vertices $V(\Gamma_H)$ correspond to cosets $Hg \subset F$ accessible by closed loops, and for each $b \in B$ there is a set $E_b(\Gamma_H)$ of directed $b$-edges, each with specified source $s(e)$ and target $t(e)$ in $V(\Gamma_H)$. The core property ensures that, except possibly at a chosen basepoint, no vertex is a leaf.

2. Definition and Properties of Polymatroids on Core Graphs

Given a (possibly disconnected) finite $B$-labeled core graph $\Gamma$, one defines a $\Gamma$-polymatroid as a system of set functions

$$h^V: 2^{V(\Gamma)} \to \mathbb{N}, \qquad h^b: 2^{E_b(\Gamma)} \to \mathbb{N}, \quad (b \in B)$$

subject to the following axioms for all $X \subseteq Y$:

• Monotonicity: $h^*(X) \leq h^*(Y)$,
• Submodularity: $h^*(X) + h^*(Y) \geq h^*(X \cup Y) + h^*(X \cap Y)$.

Moreover, the structure map between edge sets and vertex sets via source and target must preserve or dominate rank differences: $h^b(Y) - h^b(X) \geq h^V(s(Y)) - h^V(s(X))$ for all $X \subseteq Y \subseteq E_b$, and similarly for $t$. When equality holds, the polymatroid is called lossless.

This abstract formalism allows the encoding of various geometric or algebraic quantities, such as vertex sets, edge sets, covering numbers, or invariants of lifts, into a unified structure.

3. The Main Γ–Polymatroid Theorem and its Consequences

The key structural result is the Γ–polymatroid theorem. For any connected $B$-core graph $\Gamma$ with fundamental group $H = \pi_1(\Gamma)$ and $\Gamma$-polymatroid $h$, assume

• $\mathrm{rank}(H) > 1$; or
• $H$ is cyclic, $H = \langle w\rangle$ with $w$ a non-power word, and $h$ is compact (no ground element is a co-loop).

Then there exists $b \in B$ and $e \in E_b(\Gamma)$ so that

$$\chi(h) = h^V(V(\Gamma)) - \sum_{b \in B} h^b(E_b(\Gamma)) \leq -h^b(\{e\})$$

In particular, if $h^b(\{e\}) > 0$ for all $e$, then $\chi(h) < 0$.

Implications:

• For the Euler–characteristic polymatroid, this yields the Friedman–Mineyev lower bound in the strengthened Hanna Neumann inequality.
• For $h$ encoding covering counts (preimages in coverings), this reproduces Wise’s rank-1 (cyclic) bound for non-power words.
• For $h$ measuring stable primitivity rank, the theorem formalizes the Gap Theorem: $s\pi(H) \geq 1$ for non-abelian $H$, and $s\pi(w) \geq 1$ for a cyclic $H = \langle w\rangle$.

4. Probabilistic Applications: Random Homomorphisms and Invariant Sets

Let $\alpha: F \to G$ be a uniformly random homomorphism and $G$ act on a finite set $X$. Constructing polymatroids on the solution set of systems $\alpha(b).f(s(e)) = f(t(e))$, one obtains powerful upper bounds on the probability of prescribed invariant configurations.

The Reiter–Chen–Yeung type bound states: for $H$ as above, and any locally recoverable labeling $f: V(\Gamma_H) \to X$,

$|\mathcal O| \cdot \Pr_{\alpha}\bigl(\mathcal O\mbox{ is valid}\bigr) \leq \frac{1}{|X|}$

for any orbit $\mathcal O$ of $X^{V(\Gamma_H)}$. In particular, the expected number of invariant points under $\alpha(H)$ is $O(|X|^{-1})$.

Specializing, for symmetric groups $S_n$, let $S_n^H(d)$ denote the expected number of $d$-element subsets of $[n]$ fixed by $\alpha(H)$. Then for fixed $d$,

$$S_n^H(d) = \Theta\!\left(\binom{n}{d}^{-s_d(H)}\right)$$

as $n \to \infty$, with $s_d(H)$ the $d$–stable compressed rank of $H$. A similar formula holds for the Grassmannian action of $\mathrm{GL}_n(\mathbb{F}_q)$.

Problem	Bound via Polymatroid	Stable Invariant
$\lvert H \cap K \rvert$	Strengthened HN bound	Euler–characteristic
Random fixed points	$O(|X|^{-1})$	Compressed rank $s_d(H)$
Cover-lifts	Wise–type bound	Counting polymatroid

5. Worked Example: Polymatroids for a “Theta” Graph

Consider $F = \langle x, y\rangle$ and $H = \langle x y x, y x^2\rangle$. The Stallings core $\Gamma_H$ is a theta-shaped graph with two loops and one extra edge.

• The trivial polymatroid: $h^V(U) = |U|$, $h^b(E_b') = |E_b'|$ for all $U \subseteq V, E_b' \subseteq E_b$.
  • For each edge $e$, $\chi(h) = |V| - |E| = -1$ and $h^b(\{e\}) = 1$, which makes $\chi(h) \leq -1$ sharp by the theorem.
• The random-cover counting polymatroid: $h^V(U)$ and $h^b(E_b')$ defined as the logarithms of the number of partial lifts over $U$ or $E_b'$, respectively.
  • This formulation yields Wise’s bound on the number of lifts, replacing simple cardinality counts by geometric or probabilistic enumeration.

6. Open Questions and Conjectures

Several conjectures and open questions are posed based on the structure afforded by the polymatroid formalism:

• $K$–Hanna–Neumann conjecture: For any subfield $K$ and algebraic f.g. right-submodule $M \le K[F]^d$ not contained in a free summand,

  $$\operatorname{rank}_K\left(M \cap K[H]^d\right) - d \leq (\operatorname{rank}_K(M) - d)\cdot(\operatorname{rank}(H) - 1)$$

• Stability of primitivity ranks: Proposed equalities $s\pi(H) = \pi(H) - 1$ and $s\bar\pi(H) = \bar\pi(H) - 1$ suggest alignment between “stable” and one–step invariants.
• Integer-valued stable compressed ranks: Whether $s_d(H) \in \mathbb{Z}$ for all $d$, and whether $s_d(H)$ genuinely depends on $d$, remain unresolved. A conjecture posits $s_d(H) = \bar\pi(H) - 1$ for all $d$.

A plausible implication is that the minimal complexity of $d$-covers is governed by the compressed-rank lattice, indicating a deep underlying discrete structure in subgroup actions and coverings.

7. Synthesis and Unified Perspective

Polymatroids on Stallings core graphs subsume previously distinct threads: group-theoretic intersection inequalities, probabilistic behavior of group actions, enumerative invariants of coverings, and module-theoretic bounds. By encoding covering counts, fixed point decay, and linear-algebraic dependencies into polymatroid rank functions, a single framework yields gap theorems for group invariants and precise decay rates for the measure of invariant configurations under random mappings or actions. This flexible approach facilitates both the verification of classical results—such as the strengthened Hanna Neumann and Wise’s cyclic inequalities—and the formulation of new probabilistic, algebraic, and stability conjectures for subgroups and their actions (Shomroni, 31 Dec 2025).