$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, number of real and complex embeddings, etc.)?
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There is an exact sequence $\operatorname{SL}(2,R)_\text{ab}\rightarrow \operatorname{GL}(2,R)_\text{ab} \xrightarrow{\ \det\ }R^*\rightarrow 1 $. Now, if $R$ is not a ring of imaginary quadratic integers, $\operatorname{SL}(2,R)_\text{ab}\otimes \mathbb{Q} $ is zero: this follows from the Corollary to Theorem 3 in Le problème des groupes de congruence pour $\operatorname{SL}_2 $ by J.-P. Serre, Ann. Math. 92, no. 3, 489–527 (1970). Therefore $\det$ induces an isomorphism $\operatorname{GL}(2,R)_\text{ab}\otimes \mathbb{Q}\rightarrow R^*\otimes \mathbb{Q}$ in that case. You'll find how to treat the imaginary quadratic case in the same paper.
