  We give several sharp estimates for a class of combinations of second order
Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply
connected, composed of a discrete abelian component ${G}_{x}$ and a connected
component ${G}_{y}$ endowed with a biinvariant measure. These estimates include
new sharp $L^p$ estimates via Choi type constants, depending upon the
multipliers of the operator. They also include weak-type, logarithmic and
exponential estimates. We give an optimal $L^q \to L^p$ estimate as well.
  It was shown recently by Arcozzi, Domelevo and Petermichl that such second
order Riesz transforms applied to a function may be written as conditional
expectation of a simple transformation of a stochastic integral associated with
the function.
  The proofs of our theorems combine this stochastic integral representation
with a number of deep estimates for pairs of martingales under strong
differential subordination by Choi, Banuelos and Osekowski.
  When two continuous directions are available, sharpness is shown via the
laminates technique. We show that sharpness is preserved in the discrete case
using Lax-Richtmyer theorem.
