Shift maps are not type-preserving

Abstract For a surface S of sufficient complexity, Dehn twists act elliptically on the arc, curve, or relative arc graph of S . We show that composing a Dehn twist with a shift map results in a loxodromic isometry of the relative arc graph $$\mathcal {A}(S,p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for any surface S with an isolated puncture p admitting a shift map. Therefore, shift maps are not type-preserving.