Explore the intricate connections between algebraic, geometric, and analytic properties of 3-dimensional manifolds in this Rothschild Lecture delivered by Professor Cameron Gordon from the University of Texas at Austin. Delve into the fundamental group as a crucial invariant of 3-dimensional manifolds and examine how the algebraic question of left-invariant total order relates to geometric and analytic properties. Gain insights into the complex interplay between different mathematical perspectives in the study of 3-manifold groups during this hour-long talk presented at the Isaac Newton Institute for Mathematical Sciences as part of the Homology Theories in Low Dimensional Topology programme.