Involutions on sapphire Sol 3-manifolds and the Borsuk-Ulam theorem for   maps into $R^n$
  For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, \tau; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $\tau$ is a free involution, we show if $(M, \tau; R^n)$ has the Borsuk-Ulam property or not. It is known that for $n>3$ the Borsuk-Ulam property does not hold independent of the involution, so we provide a classification when $n=2$ and $3$. 
Alexandre Paiva Barreto and Daciberg Lima Gon\c{c}alves and Daniel   Vendr\'uscolo
math.AT