Groups possessing only indiscrete embeddings in SL(2,C)

TL;DR

This paper investigates conditions under which finitely generated groups, especially 3-manifold groups, have only indiscrete embeddings in SL(2,C), providing new examples and counterexamples related to embedding properties.

Contribution

It offers new criteria for when groups have only indiscrete embeddings in SL(2,C) and presents counterexamples to Minsky's simple loop question.

Findings

Certain 3-manifold groups embed in SL(2,C) depending on boundary identifications

Provided counterexamples to Minsky's simple loop question

Characterized when groups have only indiscrete embeddings in SL(2,C)

Abstract

We give results on when a finitely generated group has only indiscrete embeddings in SL(2,C), with particular reference to 3-manifold groups. For instance if we glue two copies of the figure 8 knot along its torus boundary then the fundamental group of the resulting closed 3-manifold sometimes embeds in SL(2,C) and sometimes does not, depending on the identification. We also give another quick counterexample to Minsky's simple loop question.