Semigroups of isometries of the hyperbolic plane

Jacques, Matthew and Short, Ian (2021). Semigroups of isometries of the hyperbolic plane. International Mathematics Research Notices (Early Access).



Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real Möbius transformations. We define a semigroup $S$ of Möbius transformations to be semidiscrete if the identity map is not an accumulation point of $S$. We say that $S$ is inverse free if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of Möobius transformations, then $S$ is semidiscrete and inverse free if and only if every sequence of the form $F_n=f_1\cdots f_n$, where $f_n\in\mathcal{F}$, converges pointwise on the upper half-plane to a point on the ideal boundary, where convergence is with respect to the chordal metric on the extended complex plane. We fully classify all two-generator semidiscrete semigroups, and include a version of Jörgensen's inequality for semigroups.

We also prove theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we prove that every semigroup is one of four standard types: elementary, semidiscrete, dense in the Möbius group, or composed of transformations that fix some nontrivial subinterval of the extended real line. As a consequence of this theorem, we prove that, with certain minor exceptions, a finitely-generated semigroup $S$ is semidiscrete if and only if every two-generator semigroup contained in $S$ is semidiscrete.

After this we examine the relationship between the size of the `group part' of a semigroup and the intersection of its forward and backward limit sets. In particular, we prove that if $S$ is a finitely-generated nonelementary semigroup, then $S$ is a group if and only if its two limit sets are equal.

We finish by applying some of our methods to address an open question of Yoccoz.

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