Conjugator length in finitely presented groups
The conjugator length function of a finitely generated group G gives the minimal upper bound on the length of a conjugator for a pair of words that represent conjugate elements in G, as a function of the sum of the lengths of the words. Here, we seek to promote the systematic study of conjugator length functions by explaining their significance, by surveying what is known about them and by explaining fundamental techniques and examples.
The surveyed field measures conjugacy complexity in finitely presented groups via the conjugator-length function. The formalized flagship example gives the sharp quadratic worst-case growth of shortest conjugators in the integral Heisenberg group.
Reproduction
~ partially reproduced
Attempted: Theorem 4.9, unconditional: for the standard generators of the integral Heisenberg group H₃(ℤ), the conjugator-length function satisfies CL ≃ n² under the paper's comparison relation. No hypotheses or custom axioms.
This is a survey paper — it has no single central theorem, so no run can grade it reproduced; the machine attempt formalized its flagship worked example in full. The self-contained development proves the matrix-model correspondence, standard relators, the genuine minimum word metric, equivalence with the paper's naive three-word definition, CL(n) ≤ 4n², and n² ≤ CL(4n+2), with zero sorry. Independent re-verification: all modules elaborate cleanly, kernel-checked, standard axioms only.
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