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The unique even Coxeter system minimizes exponential growth among all systems generating an even Coxeter group.

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2026-07-03 03:19 UTC pith:L7JOBQOA

load-bearing objection Even Coxeter systems minimize exponential growth among all systems for the same even group, via non-increase along Mihalik's algorithm. the 2 major comments →

arxiv 2607.01879 v1 pith:L7JOBQOA submitted 2026-07-02 math.GR

Exponential growth rates of even Coxeter groups

V\'era Bossart , Michelle Bucher This is my paper
classification math.GR MSC 20F55
keywords even Coxeter groupsexponential growth rateCoxeter systemsMihalik algorithmblow-downspseudo-transpositionsgrowth series
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for any even Coxeter group W, the even Coxeter system has the smallest exponential growth rate of any Coxeter system that generates W. The argument proceeds by tracking growth rates through Mihalik's explicit algorithm, which converts an arbitrary Coxeter system for W into the unique even one. The key step is proving that each blow-down along a pseudo-transposition leaves the growth rate unchanged or smaller. A sympathetic reader would care because the growth rate governs the asymptotic size of balls in the Cayley graph and therefore controls many geometric and combinatorial features of the group.

Core claim

Among all Coxeter systems generating an even Coxeter group W, the unique even Coxeter system realizes the minimal exponential growth. The proof compares the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one. The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.

What carries the argument

Blow-downs along pseudo-transpositions: diagram operations that replace one Coxeter system by another while preserving or decreasing the exponential growth rate of the generated group.

Load-bearing premise

Mihalik's algorithm eventually reaches the unique even system and blow-downs along pseudo-transpositions never increase growth rate.

What would settle it

Compute the exponential growth rate of a concrete even Coxeter group from a non-even Coxeter system obtained before the final blow-down and check whether it is strictly smaller than the rate of the even system.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every even Coxeter group possesses a canonical Coxeter system of minimal growth.
  • The growth rate is invariant or decreases at each step of Mihalik's reduction to the even system.
  • The minimal growth rate can be read off directly from the even Coxeter diagram.
  • Any two Coxeter systems for the same even group are connected by a sequence of moves whose growth rates are non-increasing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Minimal growth rates for even Coxeter groups can be computed by first reducing to the even diagram rather than searching over all presentations.
  • The same comparison technique might apply to growth series or other asymptotic invariants preserved by the algorithm.
  • If analogous reduction algorithms exist for other classes of groups, the minimal-growth presentation could be identified in those settings as well.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper proves that among all Coxeter systems generating an even Coxeter group W, the unique even Coxeter system realizes the minimal exponential growth rate. The argument proceeds by comparing growth rates along the steps of Mihalik's algorithm, which transforms any Coxeter system of W into the even one, with the main new ingredient being a proof that blow-downs along pseudo-transpositions do not increase the exponential growth rate.

Significance. If the new inequality on pseudo-transposition blow-downs holds in full generality, the result would canonically identify a minimal-growth Coxeter system for even groups and supply an algorithmic method to reach it. The constructive reduction via Mihalik's algorithm and the explicit non-increase claim are strengths that make the comparison falsifiable in principle.

major comments (2)
  1. [Proof of the main inequality (the new ingredient referenced in the abstract)] The global minimality claim rests entirely on non-increase of growth rate under every step of Mihalik's algorithm, including the novel case of blow-downs along pseudo-transpositions. The manuscript must verify that the spectral-radius comparison for these blow-downs remains valid when the pseudo-transposition shares vertices or edges with other generators, as the associated Coxeter matrix may then have altered off-diagonal entries that could affect the radius.
  2. [Application of Mihalik's algorithm to even diagrams] The reduction assumes Mihalik's algorithm reaches the even system from every starting Coxeter system without any intermediate step increasing growth. The paper should exhibit at least one concrete even Coxeter diagram containing a pseudo-transposition and compute the growth rates before and after the blow-down to confirm the inequality numerically.
minor comments (1)
  1. [Introduction and notation section] Notation for the growth rate (spectral radius of the Coxeter matrix) should be introduced once with a fixed symbol and used consistently; the abstract and introduction use slightly varying phrasing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the key strengths of the argument. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Proof of the main inequality (the new ingredient referenced in the abstract)] The global minimality claim rests entirely on non-increase of growth rate under every step of Mihalik's algorithm, including the novel case of blow-downs along pseudo-transpositions. The manuscript must verify that the spectral-radius comparison for these blow-downs remains valid when the pseudo-transposition shares vertices or edges with other generators, as the associated Coxeter matrix may then have altered off-diagonal entries that could affect the radius.

    Authors: The proof in Section 4 establishes the non-increase for blow-downs along pseudo-transpositions by comparing the spectral radii of the relevant Coxeter matrices in full generality. The argument proceeds from the explicit form of the growth series and does not assume that the support of the pseudo-transposition is disjoint from the remainder of the diagram; changes to off-diagonal entries are incorporated into the matrix comparison. To address the concern explicitly, we will insert a short remark immediately after the statement of the main inequality clarifying that the comparison holds without any disjointness hypothesis. revision: partial

  2. Referee: [Application of Mihalik's algorithm to even diagrams] The reduction assumes Mihalik's algorithm reaches the even system from every starting Coxeter system without any intermediate step increasing growth. The paper should exhibit at least one concrete even Coxeter diagram containing a pseudo-transposition and compute the growth rates before and after the blow-down to confirm the inequality numerically.

    Authors: We agree that a concrete numerical check would make the new inequality more transparent. In the revised manuscript we will add an explicit example of an even Coxeter diagram containing a pseudo-transposition (for instance, a small diagram on four or five generators), compute the exponential growth rates before and after the blow-down via the spectral radius of the associated matrix, and verify numerically that the rate does not increase. This example will appear in Section 5. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external algorithm plus independent inequality

full rationale

The paper establishes minimality of the even Coxeter system's growth rate by comparing rates along Mihalik's algorithm (an external reference) and proving a new inequality that blow-downs along pseudo-transpositions do not increase the rate. No self-definitional reductions, no fitted parameters renamed as predictions, and no load-bearing self-citations appear in the derivation chain. The central claim therefore rests on an independent comparison and a novel inequality rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on the correctness of Mihalik's algorithm (domain assumption) and the new blow-down inequality; no free parameters or invented entities appear in the abstract.

axioms (1)
  • domain assumption Mihalik's algorithm converts any Coxeter system of an even group into the unique even system
    Explicitly invoked as the comparison vehicle.

pith-pipeline@v0.9.1-grok · 5587 in / 981 out tokens · 28507 ms · 2026-07-03T03:19:53.350946+00:00 · methodology

0 comments
read the original abstract

Let $W$ be an even Coxeter group. We prove that among all Coxeter systems generating $W$ the unique even Coxeter system realizes the minimal exponential growth. Our proof relies on comparing the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one. The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.

discussion (0)

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Reference graph

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