Summary: Appendix to A rigidity theorem for the solvable
Baumslag­Soleter groups.
Daryl Cooper
August 18, 1997
Introduction
The study of properties of a metric space which are preserved by bilipschitz homeomorphism occurs
in the study of groups via a word metric. It was also studied in [1] for a certain type of Cantor
set embedded in the real line. The Cantor sets concerned are mild generalizations of the original
middle­third Cantor set. This Cantor set has the basic property that it is the union of two exact
copies of itself each scaled down in size by a factor of 1=3: The generalization allows finitely many
linear scale factors. It is easy to see that the Hausdorff dimension of this type of Cantor set depends
only on these scale factors. Now a bilipschitz homeomorphism preserves Hausdorff dimension, and
so a natural question is what, if any, further invariants other than Hausdorff dimension are there.
An almost complete answer was given in [1], using invariants derived from the Hausdorff measure.
This work was generalized to analogous Cantor sets in Euclidean space of dimension n by H. Vuong
in his thesis, [2], [3]. In this section, we generalize in a different direction to abstract metric Cantor
sets which possess a certain linear self­similarity structure. There is a further generalization to a
much wider class of Cantor set, where the self­similarity structure is smooth rather than linear. This
will not be dealt with here.
The main result is Theorem (0.6) which states that every bilipschitz homeomorphism between

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics