Images of Real Representations of $SL_n(Z_p)$

TL;DR

This paper studies the minimal dimensions for infinite-image representations of special linear groups over p-adic and similar rings into real general linear groups, revealing infinite minimal dimensions in positive characteristic cases.

Contribution

It determines the minimal dimension for such representations to have infinite image, especially highlighting the case of positive characteristic rings where it is infinite.

Findings

Minimal dimension for infinite image representations over p-adic rings.

Infinite minimal dimension for positive characteristic rings.

Characterization of representations over complete discrete valuation rings.

Abstract

In this paper, we investigate abstract homomorphism from the special linear group over complete discrete valuation rings with finite residue field, such as the ring of p-adic integers, into the general linear group over the reals. We find the minimal dimension in which such a representation has infinite image. For positive characteristic rings, this minimum is infinity.