From Integer Matrices to Solenoids and Odometers

Abstract: We study subgroups of Qⁿ, where Q is the field of
rational numbers, generated by integer powers of matrices with integer
entries. A central problem is to determine when two such groups are
isomorphic as abstract groups, and we show how this question can be
resolved using number-theoretic methods directly from the defining
matrices. Beyond the algebraic framework, these groups arise naturally
across mathematics: as solenoids in topology, odometers in dynamical
systems, and shifts in symbolic dynamics.