Geodesic
Teichmüller spaces of non-discrete groups
Jun Hu1 ; Francisco G. Jimenez-Lopez2
1 Brooklyn College of CUNY, Department of Mathematics, and Graduate Center of CUNY, Ph.D. Program in Mathematics, and Institute of Mathematical Sciences at NYU Shanghai
2 Universidad Autónoma de Querétaro, Facultad de Ingeniería
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 381-393.

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The concept of Teichmüller space of a Fuchsian group can be extended to any non-discrete group of conformal homeomorphisms of the hyperbolic plane. In this paper, we first present three models of Teichmüller space fulfilling that goal. Then we use them to study the Teichmüller spaces T(G) of the non-discrete subgroups G of PSL(2,R). We show that T(G) is not trivial if and only if G is a subgroup consisting of hyperbolic elements with two common fixed points and accumulating to at least one non-identity element. Furthermore, we show that if T(G) is not trivial, then (1) T(G) is conformally equivalent to the open unit disk, (2) the Teichmüller metric on T(G) is equal to the hyperbolic metric on the disk, and (3) the length spectrum is just a pseudometric on T(G) and when restricted to a one-dimensional real slice, it is a metric coinciding with the Teichmüller metric.
DOI : 10.54330/afm.114460
Keywords: Teichmüller space, length spectrum metric, non-discrete groups

Jun Hu 1 ; Francisco G. Jimenez-Lopez 2

1 Brooklyn College of CUNY, Department of Mathematics, and Graduate Center of CUNY, Ph.D. Program in Mathematics, and Institute of Mathematical Sciences at NYU Shanghai
2 Universidad Autónoma de Querétaro, Facultad de Ingeniería 
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Jun Hu; Francisco G. Jimenez-Lopez. Teichmüller spaces of non-discrete groups. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 381-393. doi : 10.54330/afm.114460. http://geodesic.mathdoc.fr/articles/10.54330/afm.114460/

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