By Ollivier Y.
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Extra info for A January invitation to random groups
We refer to [CCJJV01] for a fact sheet on the Haagerup property. For discrete groups, a very nice equivalent definition is the existence of a proper action on a space with measured walls [CMV04]. The strategy is to construct walls [HP98] in the group. Natural candidates to be walls are hypergraphs [Wis04], which are graphs built from the Cayley complex as follows: the vertices of the hypergraphs are midpoints of edges of the Cayley complex, and the edges of the hypergraphs connect vertices corresponding to midpoints of diametrally opposite edges in faces of the Cayley complex (assuming that all relators have even length).
Kazhdan’s property (T ) of a group has to do with the behavior of unitary actions of the group on the Hilbert space and basically asks that, if there are unitary vectors which the group action moves by arbitrarily small amounts, then there is a vector fixed by the action. It has proven to be linked with numerous algebraic or geometric properties of the group. We refer to [HV89, BHV, Val02a] for reviews and basic properties. t. some generating set. The neatest ˙ ˙ ´ statement is to be found in [Zuk03], see also [Zuk96, BS97, Pan98, Wan98, Val02a].
At density 0, the Cayley graph of the group is not planar [AC04] (planarity of Cayley graph and complexes is an old story, see discussion in [AC04]). The result actually holds for generic C (1/8) small cancellation groups and so: Theorem 24 – Let d < 1/16. t. the standard generating set) of a random group at density d is not planar. Actually the technique used in [AC04] allows to embed subdivisions of lots of finite graphs into the Cayley graph of a small-density random group. e. Growth exponent.