Bi-Lipschitz versus quasi-isometry equivalence
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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
Theorem 8 Any qi of is bounded distance from a height-preserving map of the form
where and are bL maps of the