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We show that being finitely presentable and being finitely presentable withsolvable word problem are quasi-isometry invariants of finitely generated leftcancellative monoids. Our main tool is an elementary, but useful, geometriccharacterisation of finite presentability for left cancellative monoids. Wealso give examples to show that this characterisation does not extend tomonoids in general, and indeed that properties such as solvable word problemare not isometry invariants for general monoids.

Quasi-isometry and finite presentations of left cancellative monoids