Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-trivial isometry group?

  • $\begingroup$ Giving conditions in terms of something like Fenchel-Nielsen coordinates looks like a difficult problem. If you just look for some example of asymmetric surfaces, then you find it in: Leon Greenberg: "Maximal groups and signatures." Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 207–226. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974. $\endgroup$ – ThiKu Oct 29 '15 at 17:38
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    $\begingroup$ What kind of conditions do you want? (that is, in terms of what invariants)? $\endgroup$ – Igor Rivin Oct 29 '15 at 18:42
  • $\begingroup$ @IgorRivin To be honest, I am not entirely sure, because I am merely curious and don't have a very specific reason for asking this. Vaguely, I am looking for some kind of heuristic which says that "many" hyperbolic metrics are highly symmetric, or may be something entirely opposite: the hyperbolic metrics that have any sort of non-trivial symmetry have measure zero. $\endgroup$ – user82102 Oct 29 '15 at 19:04
  • $\begingroup$ @IgorRivin In the latter situation, it would also be nice to get an insight into what obstructs the generic hyperbolic metric from having any symmetry. $\endgroup$ – user82102 Oct 29 '15 at 19:07

In genus 2, every surface has a symmetry, namely a hyperelliptic involution. In higher genus, generic surfaces will not have any symmetries. If a surface has a non-trivial symmetry group, then the quotient by the symmetry group will be an orbifold, and the moduli space of hyperbolic structures on this orbifold will have strictly smaller dimension. Hence moduli space will not be covered by these smaller dimensional Teichmüller spaces.

One possible obstruction: suppose a surface admits a symmetry (not hyperelliptic in genus 2), then there will be simple closed geodesics which get moved by an isometry, and hence there are simple closed curves which have the same lengths. So if a surface has all simple closed curves of different lengths, then it cannot admit any non-trivial symmetries. One could probably also state this for curves of some bounded length depending on the genus and injectivity radius.

To see that some curve must be sent to a distinct curve, suppose not. Then every non-separating curve intersects another curve in a single point. Then this point must be fixed by the isometry, and in fact the isometry must be an involution fixing this point. Arguing like this, one can conclude that the isometry is a hyperelliptic involution. However, in genus $>2$, there are always curves not fixed by any given hyperelliptic.

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    $\begingroup$ I should add that such surfaces exist. For any pair of embedded curves, the set of surfaces in which they have the same length is a codimension one subset. So by Baire category, there exist surfaces in which all of the curves have different lengths. This is not true for immersed curves though, for which certain geodesics will always have the same length. $\endgroup$ – Ian Agol Oct 30 '15 at 17:32

I'll just point out an answer based on somewhat different criteria than explicitly knowing features of the hyperbolic metric: As is well-known, every oriented, compact hyperbolic surface $C$ is canonically a Riemann surface of genus $g\ge2$.

As Ian points out, if $C$ is hyperelliptic, it always has a symmetry, the hyperelliptic involution $\iota:C\to C$, and it can be written, almost canonically, as an algebraic plane curve $$ y^2 = (x-\lambda_1)\cdots(x-\lambda_{2g+2}), $$ where the $\lambda_i$ are distinct complex numbers. Let $\Gamma\subset\mathrm{PSL}(2,\mathbb{C})$ be the (finite) group of linear fractional transformations that preserve the set $\Lambda =\{\lambda_1,\ldots,\lambda_{2g+2}\}$. Then the group of orientation preserving isometries of $C$ will be an extension of $\Gamma$ by a $\mathbb{Z}_2$ (because of the hyperelliptic involution). If lets $\Gamma_+$ be the (possibly slightly larger) group of $\pm$-holomorphic linear fractional transformations of $\mathbb{P}^1$ that preserve $\Lambda$, then the full group of isometries of $C$ will be a $\mathbb{Z}_2$ extension of $\Gamma_+$.

If $C$ is not hyperelliptic, then its canonical mapping $\kappa:C\to\mathbb{CP}^{g-1}$ is an embedding, and the isometries of $C$ are exactly the $\pm$-holomorphic automorphisms of $\mathbb{CP}^{g-1}$ that preserve the image $\kappa(C)$. For example, when $g=3$ and $C$ is not hyperelliptic, the image $\kappa(C)\subset\mathbb{CP}^2$ is a nonsingular plane quartic $Q(X,Y,Z)=0$, and every nonsingular plane quartic is a canonical curve of genus $3$. The orientation-preserving isometries of $C$ are exactly the elements of $\mathrm{PSL}(3,\mathbb{C})$ that preserve the quartic $Q$. Since one can write down explicit homogeneous nonsingular quartics in $3$ variables that have no nontrivial automorphisms, it follows that the generic curve of genus $3$ has no nontrivial automorphisms, and hence the conformal hyperbolic metric on $C$ has no nontrivial isometries.

This gives one a way to write down 'explicit' examples, modulo, of course, the fact that there is no known way explicitly to 'write down' the conformal hyperbolic metric on a nonhyperelliptic curve of genus $3$.


Let me add to this nice answer of Ian Agol, that the isometry group is always finite, and contains at most $84(g-1)$ elements, if we count orientation-preserving (conformal) isometries, where $g\geq 2$ is the genus (Hurwitz, Math Ann, 41 (1893)). For an exposition in English, see the book by Tsuji, Potential theory in modern function theory.

EDIT. Let me also mention the beautiful book, The Eightfold Way (S. Levy, editor) which contains several surveys on the subject, in particular Hurwitz grous and surfaces, by Murray Macbeath, where he describes history and motivation for this theorem.

  • $\begingroup$ That's pretty interesting. Is it known whether the bounds are attained? And at the risk of sounding very vague, is it known what kind of groups could actually be isometry groups of hyperbolic surfaces? $\endgroup$ – HSM Nov 13 '15 at 23:03
  • $\begingroup$ The bounds are attained (see the references that I included). I believe that the second question is addressed in the second reference (S. Levy). $\endgroup$ – Alexandre Eremenko Nov 14 '15 at 0:21
  • $\begingroup$ Also see the Wikipedia article "Hurwitz's automorphisms theorem". It has a nice discussion of the subject. $\endgroup$ – Alexandre Eremenko Nov 14 '15 at 0:24

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