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Let  denote the Markov type  constant at time  of a metric space , where . We show that  in each of the following cases: (a)  and  are geodesic spaces and  is covered by  via a finite-sheeted locally isometric covering, (b)  is the quotient of  by a finite group of isometries, (c)  is the -Wasserstein space over . As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the -Wasserstein space over . These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.

Markov type constants, flat tori and Wasserstein spaces