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Word length versus lower central series depth for surface groups and RAAGs

Justin Malestein, Andrew Putman

Abstract

For surface groups and right-angled Artin groups, we prove lower bounds on the shortest word in the generators representing a nontrivial element of the kth term of the lower central series.

Word length versus lower central series depth for surface groups and RAAGs

Abstract

For surface groups and right-angled Artin groups, we prove lower bounds on the shortest word in the generators representing a nontrivial element of the kth term of the lower central series.
Paper Structure (25 sections, 5 theorems, 32 equations)

This paper contains 25 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Free groups
  3. Upper bounds
  4. Surface groups
  5. Right-angled Artin groups
  6. From RAAGs to surface groups
  7. Optimal embeddings
  8. Sublinearity
  9. Outline
  10. Acknowledgments
  11. Right-angled Artin groups
  12. Monoid
  13. Normal form
  14. Monoid ring
  15. Partially commuting power series
  16. ...and 10 more sections

Key Result

Theorem A

Let $\pi$ be a nonabelian surface group with standard generating set $S = \{a_1,b_1,\ldots,a_g,b_g\}$. Then for all $k \geq 1$ we have $d_{\pi,S}(k) \geq \frac{1}{4} k$.

Theorems & Definitions (17)

  • Remark 1.1
  • Theorem A
  • Conjecture 1.2
  • Example 1.3
  • Remark 1.4
  • Theorem B
  • Remark 1.5
  • Conjecture 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 7 more