** 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, is qi to but not bL to .

Say a subset of a metric space is a *Delaunay set* if it is uniformly discrete and coarsely dense.

**Exercise**. Every Delaunay set of is bL to .

**Theorem 1 (Burago-Kleiner, McMullen 1998)** * If , there exist Delaunay sets in not bL to . *

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

**Theorem 3 (Dymarz-Navas 2016)** * Idem in , in . *

**Question**. In such groups, are there non-bL lattices? Answer is negative for , and . 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 in is , for any finite subset , such a bijection would map tubular neigborhood onto , hence

which implies a linear isoperimetric inequality.

However, there is a bL map of to : precompose injection with . Indeed, this map is a 2-1 qi.

In general, if admits a self-qi, composition with an index inclusion is bounded distance from a bL map.

** 2.3. When does a group admit a 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 where . Then has index in .

**Theorem 7 (Dymarz)** * and are not bL if is not a product of primes appearing in . *

Start with a -regular tree , with a height function . Then the Cayley graph of has vertices the set of pairs

An edge

**Theorem 8** * Any qi of is bounded distance from a height-preserving map of the form *

*
** where and are bL maps of the *

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metric2011 is a program of Centre Emile Borel, an activity of Institut Henri PoincarĂ©, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/