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The infinite dimensional geometry of conjugation invariant generating sets

Sabine Chu, George Domat, Christine Gao, Ananya Prasanna, Alex Wright

Abstract

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all non-filling curves; mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements; and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on work of Brandenbursky-Gal-Kȩdra-Marcinkowski, in these Cayley graphs we show that there are quasi-isometrically embedded copies of $\mathbb{Z}^m$ for all $m\geq 1$. A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka-Savchuk's result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of $\mathbb{Z}^m$ for all $m\geq 1$.

The infinite dimensional geometry of conjugation invariant generating sets

Abstract

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all non-filling curves; mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements; and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on work of Brandenbursky-Gal-Kȩdra-Marcinkowski, in these Cayley graphs we show that there are quasi-isometrically embedded copies of for all . A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka-Savchuk's result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of for all .

Paper Structure

This paper contains 23 sections, 23 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Abelian behavior for conjugation invariant generating sets
  3. From quasi-morphisms to quasi-flats
  4. Quasi-morphisms coming from hyperbolic actions
  5. Free groups
  6. The quotient by conjugation
  7. The edge splitting graph
  8. Previous work finding flats of arbitrary dimension
  9. Final remarks
  10. Acknowledgments
  11. Free groups
  12. Background
  13. Many quasi-morphisms
  14. The quotient by conjugation
  15. The free-splitting graph
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $S$ be a conjugation invariant generating set for a group $G$. Suppose $G$ has $m$ linearly independent homogeneous quasi-morphisms that are bounded on $S$. Then there is a quasi-isometric embedding of $\mathbb{Z}^m$ into $\mathop{\mathrm{Cay}}\nolimits(G,S)$. $\blacktriangleleft$$\blacktriangle

Figures (3)

  • Figure 1: An illustration of the difference between $\mathop{\mathrm{Wh}}\nolimits(w)$ and $\mathop{\mathrm{Wh}}\nolimits'(w)$ for $w = a_1 a_2^2 a_3^2 a_4^2 \overline{a}_1 \in F_{4}$.
  • Figure 2: An illustration of $\mathop{\mathrm{Wh}}\nolimits(p_k)=\mathop{\mathrm{Wh}}\nolimits'(p_k)$
  • Figure 3: A fundamental domain for the axis of $a_n$ in the splitting $H \ast \langle wa_n\rangle$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 37 more