Stable Invariants of Words from Random Matrices

TL;DR

This work introduces and develops a family of stable, topology‑inspired word invariants obtained from w‑random matrices across several compact group families, extending the classical stable commutator length scl. It shows that scl is recovered from w‑measures on U(•) via stable polynomial irreps, and proposes analogous invariants (sπ, sql, sπ^{(m)}, sπ_q) tied to S_•, wreath products, and GL_•(q). The paper provides substantial results for specific families (notably U(•) and C_m ≀ S_•), proves rationality and gap properties, and formulates conjectures that these invariants capture intrinsic topological/combinatorial data of words, with several results shown to be profinite. It also outlines a broad, interconnected program linking word measures, representation theory, and topology, and raises open questions about universality, extensions, and computability of these invariants.

Abstract

Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients of $w$-random unitary matrices [arXiv:1802.04862]. But the random-matrix side of this equality can be naturally tweaked by considering $w$-random permutations, $w$-random orthogonal matrices and so on, to produce new invariants for any given word. Are these invariants new? interesting? Do they admit an intrinsic topological description as in the case of $w$-random unitaries and scl? The current paper formalizes the definition of these invariants coming from $w$-random matrices, answers the above questions in certain cases involving generalized symmetric groups, and poses detailed conjectures in many others. In particular, we present a plethora of topological, combinatorial and algebraic invariants of words which play, or are at least conjectured to play, a similar role to the one played by scl in the above-mentioned result. Among others, these invariants include two invariants recently defined by Wilton [arXiv:2210.09853]: the stable primitivity rank and a non-oriented analog of scl.

Figures (6)

• Figure 1.1: Known inequalities between some of the stable invariants discussed in this paper. In every line in the diagram, the upper invariant is known to be equal to or larger than the bottom one for every word. A broken line means that there are intermediate stable invariants bounded between the one in the bottom and the one in the top.
• Figure 2.1: On the left is a morphism from $P$, a union of two cycles, to $\Delta$, a barbell-shaped graph. The edges of $\Delta$ are oriented and labeled in order to depict the morphism: the image of every edge of $P$ is marked using these labels and orientations. The two Whitehead graph of this morphism, one for each vertex of $\Delta$, are drawn on the right. In each of the two Whitehead graphs, each vertex is labeled by the corresponding half-edge of $\Delta$ emanating from it.
• Figure 2.2: Folding and unfolding: consider an immersion of finite graphs $b\colon P\to\Delta$, where $P$ is a union of cycles. Let $v\in V\left(\Delta\right)$ and assume that $\mathrm{Wh} _{b}\left(v\right)$ contains a cut vertex $e_{0}\in\iota_{\Delta}^{-1}\left(v\right)$. Denote $u=\iota_{\Delta}\left(\overline{e_{0}}\right)$. On the left, we draw the vertex $v$ together with its incident half-edges, with the edges of $\Delta$ drawn as thick ribbons. The red and blue lines represent pieces of the $b$-image of $P$ that traverse $v$. Unfolding at $e_{0}$ results in the picture on the right. Conversely, a homotopy-equivalent folding of $\overline{e_{1}}$ and $\overline{e_{2}}$ on the right picture results in the left picture where the merged vertex admits a cut vertex.
• Figure 5.1: A degree-2 $\mathrm{ssql}$-extremal surface for $w=a^{2}b^{2}ab^{-1},$ showing that $\mathrm{ssql}\left(w\right)=1$.
• Figure 6.1: Consider $w=a^{3}ba^{-1}b^{-1}$. The upper left graph is $\Gamma_{w}$, and the bottom left one is (the decorated) $\mathrm{Wh} _{\eta_{w}}(o)$ (the black oriented edge labeled by $a$ or $b$ at every vertex only marks the corresponding half-edge of $\Omega$, and is not a genuine part of the Whitehead graph). The other four graphs depict the four types of pieces which are valid for this word when $m=3$. Here we restrict to pieces representing connected Whitehead graphs only (see Footnote \ref{['fn:only connected pieces']}).

Theorems & Definitions (100)

• Theorem 1.1
• Theorem : Theorems \ref{['thm:result on wreath products and modulo-m spi']}, \ref{['thm:rationality of spm']} and \ref{['thm:spm avoids (0,1)']}
• Theorem
• Theorem
• Conjecture 1.2
• Definition 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.5
• Corollary 1.6