Homology of symplectic groups in the unstable range

Question

Let $Sp(2n,{\mathbb R})$ be the symplectic group and $H_3(Sp(2n,{\mathbb R});{\mathbb Z})$ its 3rd group homology (i.e., for the group with the discrete topology).

It is known that $$H_3(Sp(2n,{\mathbb R});{\mathbb Z})\to H_3(Sp(2n+2,{\mathbb R});{\mathbb Z})$$ is an isomorphism for $n\ge 3$. ( Theorem 3.9 in https://arxiv.org/pdf/0905.0071v5.pdf )

My question is about the unstable range: has someone described the cokernel of $$H_3(Sp(2,{\mathbb R});{\mathbb Z})\to H_3(Sp(4,{\mathbb R});{\mathbb Z}) ?$$

Answer 1

The optimal stabilization range for third homology of the symplectic groups has been determined by Marco Schlichting and Husney Parvez Sarwar in https://arxiv.org/abs/2111.01539.

Their result states that for any local ring $R$ with infinite residue field, the stabilization map $H_3({\rm Sp}_2(R))\to H_3({\rm Sp}_4(R))$ is surjective and for $k\geq 2$ the stabilization maps $H_3({\rm Sp}_{2k}(R))\to H_3({\rm Sp}_{2k+2}(R))$ are isomorphisms, i.e., third homology of ${\rm Sp}_4(R)$ is already stable.