The R${}_{\mathrm{\infty }}$-property for braid groups over orientable surfaces

Let ${\mathrm{\Sigma }}_{g,p}$ be an orientable surface of genus $g$ and of finite type without boundary (i.e. an orientable closed surface with a finite number $p$ of points removed). In this paper we study the R${}_{\mathrm{\infty }}$-property for the surface pure braid groups $P_n({\mathrm{\Sigma }}_{g,p})$ as well as for the full surface braid groups $B_n({\mathrm{\Sigma }}_{g,p})$. We show that, with few exceptions, these groups have the R${}_{\mathrm{\infty }}$-property.