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

Bi-Lipschitz versus quasi-isometry equivalence

Advertisement for quasi-isometric rigidity shool, Toulouse, june 26th-july 14th, 2017.

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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