Thurston boundary of Teichmüller spaces and the commensurability modular group (1999)

If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p colon upper Y right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p : Y \rightarrow X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, from the Teichmüller space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> actually extends to an embedding between the Thurston compactification of the two Teichmüller spaces. Using this result, an inductive limit of Thurston compactified Teichmüller spaces has been constructed, where the index for the inductive limit runs over all possible finite unramified coverings of a fixed compact oriented surface of genus at least two. This inductive limit contains the inductive limit of Teichmüller spaces, constructed by I. Biswas, S. Nag and D. Sullivan, Determinant bundles, Quillen metrics and Mumford isomorphisms over the Universal Commensurability Teichmüller Space, <italic>Acta Mathematica</italic>, 176 (1996), 145–169, as a subset. The universal commensurability modular group, which was constructed in the above mentioned article, has a natural action on the inductive limit of Teichmüller spaces. It is proved here that this action of the universal commensurability modular group extends continuously to the inductive limit of Thurston compactified Teichmüller spaces.