Dimensions of Higher Extensions for SL_2

TL;DR

This paper investigates the growth of Ext groups for SL_2 over fields of characteristic p, revealing exponential growth in cohomology dimensions and providing generating functions for these groups.

Contribution

It derives recursive formulas and generating functions for Ext groups of SL_2, demonstrating exponential growth in cohomology dimensions.

Findings

Cohomology of SL_2 grows at least exponentially.

Max Ext^i groups for fixed i are bounded.

Generated functions describe Ext group dimensions.

Abstract

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.