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Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer

Hyungryul Baik

TL;DR

This work proves a sharp length inequality for products of conjugates equal to the identity in random groups at density $d<1/2$: for any $n\ge1$ and $\varepsilon>0$, any tight word $W=\prod_{i=1}^n h_i^{-1} g h_i$ with $g\neq 1$ satisfies $\sum_{i=1}^n |h_i| > \frac{1-2d-\varepsilon}{2}L - \frac{n}{2}|g|$ w.o.p. as $L\to\infty$, proven via van Kampen diagrams and Ollivier's sharp isoperimetric inequality. This yields uniform short-witness exclusions and a width–length tradeoff for generalized torsion at every density $d<1/2$, and, via the first-order transfer theorem of Kharlampovich–Miasnikov–Sklinos, a corollary that random groups have no generalized torsion of any fixed width. The results provide quantitative obstructions to generalized torsion in random groups and connect isoperimetric techniques with logical transfer principles to obtain both constructive and qualitative constraints.

Abstract

Let $G$ be a random group in Gromov's density model $G(m,d,L)$ with $d<\tfrac12$. We prove a sharp quantitative constraint on products of conjugates equal to the identity: for every $n\ge1$ and $\varepsilon>0$, with overwhelming probability as $L\to\infty$, any tight word \[ W=\prod_{i=1}^n h_i^{-1} g h_i =1 \quad\text{in } G \] (with $g\neq 1$ as a word) satisfies the inequality \[ \sum_{i=1}^n \len{h_i} \;>\; \frac{1-2d-\varepsilon}{2}\,L \;-\; \frac{n}{2}\,\len{g}. \] The proof is a short van Kampen diagram argument: Ollivier's sharp isoperimetric inequality forces a 2-cell contributing a large portion of its boundary to the outer boundary, and a simple boundary block-counting estimate yields this corridor-type lower bound. As consequences we obtain uniform short-witness exclusions and width--length tradeoffs for generalized torsion at every density $d<\tfrac12$. We also deduce that random groups have no generalized torsion of any fixed width as a corollary of the recent first-order transfer theorem of Kharlampovich, Miasnikov, and Sklinos.

Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer

TL;DR

This work proves a sharp length inequality for products of conjugates equal to the identity in random groups at density : for any and , any tight word with satisfies w.o.p. as , proven via van Kampen diagrams and Ollivier's sharp isoperimetric inequality. This yields uniform short-witness exclusions and a width–length tradeoff for generalized torsion at every density , and, via the first-order transfer theorem of Kharlampovich–Miasnikov–Sklinos, a corollary that random groups have no generalized torsion of any fixed width. The results provide quantitative obstructions to generalized torsion in random groups and connect isoperimetric techniques with logical transfer principles to obtain both constructive and qualitative constraints.

Abstract

Let be a random group in Gromov's density model with . We prove a sharp quantitative constraint on products of conjugates equal to the identity: for every and , with overwhelming probability as , any tight word (with as a word) satisfies the inequality The proof is a short van Kampen diagram argument: Ollivier's sharp isoperimetric inequality forces a 2-cell contributing a large portion of its boundary to the outer boundary, and a simple boundary block-counting estimate yields this corridor-type lower bound. As consequences we obtain uniform short-witness exclusions and width--length tradeoffs for generalized torsion at every density . We also deduce that random groups have no generalized torsion of any fixed width as a corollary of the recent first-order transfer theorem of Kharlampovich, Miasnikov, and Sklinos.
Paper Structure (4 sections, 10 theorems, 19 equations)

This paper contains 4 sections, 10 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. A general estimate from a large boundary face
  3. Random groups at density $d<1/2$ (proof of Theorem \ref{['thm:main']})
  4. First-order transfer and a fixed-width corollary

Key Result

Theorem 1.1

Fix $d<1/2$, $n\ge 1$, and $\varepsilon>0$. In the density model $G(m,d,L)$, w.o.p. as $L\to\infty$, the following holds. If a tight word (with $g\neq 1$ as a word, and no cancellation across parentheses after cyclic rotation) represents $1$ in the random group, then

Theorems & Definitions (25)

  • Theorem 1.1: Sharp length inequality at density $d<1/2$
  • Definition 2.1: Tight conjugate normal form
  • Remark 2.2
  • Lemma 2.3: Lower bound from one large boundary face
  • proof
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6: A large boundary face from strict linear isoperimetry
  • proof
  • Theorem 2.7: Inequality from strict linear isoperimetry
  • ...and 15 more