Stabilizers of simple paths in the Bruhat-Tits tree of SL(2) over finite extensions of Q2

TL;DR

This paper investigates the stabilizers of simple paths in the Bruhat-Tits tree of SL(2) over finite extensions of Q2, providing explicit computations of stabilizers for trace-zero matrices.

Contribution

It introduces a method to compute stabilizers of matrices in SL(2) over local fields by analyzing paths in the Bruhat-Tits tree, extending understanding of group actions.

Findings

Explicit stabilizer formulas for matrices over finite extensions of Q2

Characterization of Galois-invariant paths in the Bruhat-Tits tree

Connection between matrix conjugacy classes and geometric paths

Abstract

For F an algebraic extension of Q2, the conjugacy classes of invertible, 2-by-2, trace-zero matrices under the action of G := SL2(F) are analyzed relative to the quadratic extension that splits the respective characteristic polynomial. The stabilizer in G of each such matrix is computed as a stabilizer of a simple, Galois invariant path in the Bruhat-Tits Tree of G.