Canonicality of Makanin-Razborov Diagrams - Counterexample

TL;DR

This paper demonstrates that the canonical Makanin-Razborov diagrams for finitely generated groups depend on the chosen generating set, providing a counterexample to their invariance.

Contribution

It shows that the construction of Makanin-Razborov diagrams is not invariant under different generating sets, challenging previous assumptions about their canonical nature.

Findings

The diagram construction varies with different generating sets.

A specific counterexample illustrates the dependence.

The canonicality of the diagrams is not absolute.

Abstract

Sets of solutions to finite systems of equations in a free group, are equivalent to sets of homomorphisms from a fixed f.p. group into a free group. The latter can be encoded in a diagram, the construction of which is valid also for f.g. groups. The diagram is known to be canonical for a fixed f.g. group with a fixed generating set. In this paper we prove that the construction depends on the chosen generating set of the given f.g. group.