Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Vincent Lafforgue, Assaf Naor

Let $\stackrel{˝}{=}<a,b|a[a,b]=[a,b]a\wedge b[a,b]=[a,b]b>$ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(\cdot ,\cdot )$ associated to the generating…