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A group of paths in ℝ²

A group of paths in ℝ²

We define a group structure on the set of compact “minimal” paths in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We classify all finitely generated subgroups of this group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper …