Notes of Tullia Dymarz’ Oxford lecture 21-03-2017

Bi-Lipschitz versus quasi-isometry equivalence

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1. Examples

Different choices of finite generating systems leads to bi-Lipschitz (write bL) equivalences of a group. However, ${{\mathbb Z}}$ is qi to ${{\mathbb R}}$ but not bL to ${{\mathbb R}}$.

Say a subset ${X\subset Y}$ of a metric space is a Delaunay set if it is uniformly discrete and coarsely dense.

Exercise. Every Delaunay set of ${{\mathbb R}}$ is bL to ${{\mathbb Z}}$.

Theorem 1 (Burago-Kleiner, McMullen 1998) If ${n\geq 2}$, there exist Delaunay sets in ${{\mathbb R}^n}$ not bL to ${{\mathbb Z}^n}$.

Theorem 2 (Dymarz-Kelly-Li-Lukianenko 2016) Idem in nilpotent Lie groups.

Theorem 3 (Dymarz-Navas 2016) Idem in ${SOL}$, in ${BS(1,m)=\langle a,t|tat^{-1}=a^m\rangle}$.

Question. In such groups, are there non-bL lattices? Answer is negative for ${{\mathbb R}^n}$, ${SOL}$ and ${BS(1,m)}$. Open for non-abelian nilpotent groups.

2. When are two finitely generated groups qi but not bL ?

2.1. The non-amenable case

Theorem 4 (Whyte 1999) Any quasi-isometry between non-amenable groups is bounded distance from a bi-Lipschitz map.

The argument is purely geometric: it applies to uniformly discrete bounded geometry metric spaces.

2.2. The amenable case

Theorem 5 (Dymara 2010, Dymara-Peng-Tabuck 2015) Certain lampligter groups are qi but not bL.

Whyte’s argument definitely does not generalize:

Proposition 6 For amenable groups, finite index inclusions are not bounded distance from any bL equivalence.

Indeed, if index of ${H}$ in ${G}$ is ${k}$, for any finite subset ${S\subset G}$, such a bijection would map tubular neigborhood ${S_R\cap H}$ onto ${S}$, hence

$\displaystyle \begin{array}{rcl} (1-\frac{1}{k})|S\setminus S\cap H|\leq |S_R\setminus S|, \end{array}$

which implies a linear isoperimetric inequality.

However, there is a bL map of ${{\mathbb Z}}$ to ${{\mathbb Z}\oplus{\mathbb Z}/2{\mathbb Z}}$: precompose injection with ${x\mapsto\lfloor x/2\rfloor}$. Indeed, this map is a 2-1 qi.

In general, if ${H}$ admits a ${k-1}$ self-qi, composition with an index ${k}$ inclusion is bounded distance from a bL map.

2.3. When does a group admit a ${k-1}$ qi ?

Strongly qi-rigid (i.e. when every qi is bounded distance from a qi) amenable groups do not. Unfortunately, no example of a qu-rigid amenable group is known.

3. Lamplighter groups

Let ${G=F\wr {\mathbb Z}}$ where ${|F|=n}$. Then ${H=F^k\wr{\mathbb Z}}$ has index ${k}$ in ${G}$.

Theorem 7 (Dymarz) ${G}$ and ${H}$ are not bL if ${k}$ is not a product of primes appearing in ${n}$.

Start with a ${n+1}$-regular tree ${T}$, with a height function ${ht:T\rightarrow{\mathbb Z}}$. Then the Cayley graph of ${G}$ has vertices the set of pairs

$\displaystyle \begin{array}{rcl} \{(p,q)\in T\times T\,;\, ht(p)+ht(q)=0\}. \end{array}$

An edge

Theorem 8 Any qi of ${F\wr{\mathbb Z}}$ is bounded distance from a height-preserving map of the form

$\displaystyle \begin{array}{rcl} F(x,y,z)=(f_1(x),f_2(y),z) \end{array}$

where ${f_1}$ and ${f_2}$ are bL maps of the

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