Dimensions of Higher Extensions for SL_2

We analyse the recursive formula found for various Ext groups for ${\text{SL}}_2(k)$, $k$ a field of characteristic $p$, and derive various generating functions for these groups. We use this to show that the growth rate for the cohomology of ${\text{SL}}_2(k)$ is at least exponential. In particular, $\max {\dim \Ext^i_{\SL_2(k)}(k, \Delta(a))\mid a,i \in \N }$ has (at least) exponential growth for all $p$. We also show that $\max {\dim \Ext^i_{\SL_2(k)}(k, \Delta(a))\mid a\in \N }$forafixed$i$ is bounded.