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In this work, we solve the problem explicitly stated at the end of a paper ofJunge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groupsincluding all amenable groups and free groups. More precisely, we prove thatthe Hodge-Dirac operator of the canonical "hidden" noncommutative geometryassociated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers isbisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on abisector which implies a positive answer to the quoted problem. Our result canbe seen as a strengthening of the dimension free estimates of Riesz transformsof the above authors and also allows us to provide Hodge decompositions. A partof our proof relies on a new transference argument between multipliers which isof independent interest. Our results are even new for the Poisson semigroup on$\mathbb{T}^n$. We also provide a similar result for Markov semigroups of Schurmultipliers and dimension free estimates for noncommutative Riesz transformsassociated with these semigroups. Along the way, we also obtain new Khintchinetype equivalences for $q$-Gaussians in $\mathrm{L}^p$-spaces associated tocrossed products. Our results allow us to introduce new spectral triples (i.e.noncommutative manifolds) and new quantum (locally) compact metric spaces, inconnection with the carr\'e du champ, which summarize the underlying geometryof our setting. Finally, our examples lead us to introduce a Banach spacevariant of the notion of spectral triple suitable for our context.

Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I