Geometry Colloquium

Seminar information archive ~06/22Next seminarFuture seminars 06/23~

Date, time & place Friday 10:00 - 11:30 126Room #126 (Graduate School of Math. Sci. Bldg.)

2012/10/17

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Masato Mimura (Tohoku University)
p-Kazhdan constants and non-expanders (JAPANESE)
[ Abstract ]
In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.
In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.