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We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b.
We give an estimate for the number of closed loops of given length in the 1-skeleton of a thick euclidean building. This kind of estimate can be used to prove the (RD) property for the subspace of radial functions on groups, as shown in the paper by A. Valette [same issue].
We introduce a family of conditions on a simplicial complex that we call local -largeness (≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional...
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