The Structure of the Three-Dimensional Special Linear Group over a Local Field

TL;DR

This paper demonstrates that the group SL3(K) over a local field acts on a specific simplicial complex, revealing its structure as an amalgamated product of subgroups, generalizing known results for SL2(K).

Contribution

It extends the understanding of the structure of SL3(K) over local fields by showing it as an amalgamated product, generalizing Ihara's theorem for SL2(K).

Findings

SL3(K) acts on a 2D simplicial complex with a fundamental domain.

SL3(K) is the amalgamation of three subgroups isomorphic to SL3(O).

Generalizes Ihara's theorem for SL2(K).

Abstract

For K a local field, it is shown that SL3(K) acts on a simply connected two dimensional simplicial complex in which a single face serves as a fundamental domain. From this it follows that SL3(K) is the generalized amalgamated product of three subgroups. Specifically if K is the field of fractions of the discrete valuation ring O, then SL3(K) is the amalgamation of three subgroups isomorphic to SL3(O) along pairwise intersections. This generalizes a theorem of Ihara, which gives the structure of SL2(K) as the amalgamated product of two groups in analogous fashion.