Mathematics – Metric Geometry
Scientific paper
2007-05-31
Mathematics
Metric Geometry
Scientific paper
We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${\rm O}(\sqrt{\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \bwr C_n$ is $\Theta(\sqrt{\log n})$. Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin \cite{AhaMauMit} and by Gromov (see \cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.
Austin Tim
Naor Assaf
Valette Alain
No associations
LandOfFree
The Euclidean distortion of the lamplighter group does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Euclidean distortion of the lamplighter group, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Euclidean distortion of the lamplighter group will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-289387