The full spectrum of scl on recursively presented groups

Abstract

We show that the set $SC{L}^{rp}$ of stable commutator lengths on recursively presented groups equals the set of non-negative right-computable numbers. Hence all non-negative algebraic or computable numbers are in $SC{L}^{rp}$ and $SC{L}^{rp}$ is not closed under subtraction. We also show that every non-negative real number is the stable commutator length of an element in some infinitely presented small cancellation group.

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