 Abelianization of a semidirect product by the integers

ID: abelianization-of-a-semidirect-product-by-the-integers

Let

B \in GL_{d} (Z)

and

Γ_{B} = Z^{d} ⋊_{B} Z

. If

t

generates the second factor, then the elements

t n t^{- 1} n^{- 1}

fill

(B - I) Z^{d}

, and hence

Γ_{B}^{ab} ≅ Z \oplus Z^{d} / (B - I) Z^{d} .

1

is not an eigenvalue of

B

, the second summand is finite.

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