Dimensions of Higher Extensions for SL₂

We analyse the recursive formula found for various Ext groups for $\SL_2(k)$, $k$ 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 $\SL_2(k)$ 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 $\max \{\dim \Ext^i_{\SL_2(k)}(k, \Delta(a))\mid a\in \N \}$ for a fixed $i$ is bounded.