Infinite generation of circle stabilizers in Kleinian groups

For a Kleinian group $\Gamma$ (a discrete group of orientation-preserving isometries of hyperbolic 3-space), the stabilizer $\Gamma^C$ of a circle $C$ at infinity is a discrete group of isometries of the hyperbolic plane bounded by $C$; it is the fundamental group of the intrinsic surface of any geodesic plane immersed in $M=\Gamma\backslash\mathbb{H}^3$ whose boundary is $C$. After classifying geodesic planes in rigid acylindrical hyperbolic 3-manifolds, McMullen, Mohammadi and Oh asked in 2017 what can be said in general about such stabilizers — in particular whether $\Gamma^C$ can be infinitely generated — noting that infinitely generated groups do arise in the analogous setting for Teichmüller space. Subsequent work establishes that $\Gamma^C$ is finitely generated whenever the circle orbit $\Gamma\cdot C$ is discrete (properly immersed planes) and when $C\subset\Lambda$ (planes in the convex core, which are closed with finite area), but the question remains open in its substantive range: circles with non-discrete (exotic or dense) orbits in geometrically finite manifolds, and arbitrary circles in degenerate (geometrically infinite) manifolds. No example of an infinitely generated circle stabilizer inside a finitely generated Kleinian group is known, and no general finiteness theorem excludes one.

Background

Hyperbolic 3-space $\mathbb{H}^3$ is the unique simply connected complete Riemannian 3-manifold of constant curvature $-1$; its group of orientation-preserving isometries is $\mathrm{PSL}_2(\mathbb{C})$, and its sphere at infinity is $S^2=\partial\mathbb{H}^3$. A Kleinian group is a discrete subgroup $\Gamma\subset\mathrm{PSL}_2(\mathbb{C})$; the quotient $M=\Gamma\backslash\mathbb{H}^3$ is a hyperbolic 3-manifold (an orbifold if $\Gamma$ has torsion). The limit set $\Lambda\subset S^2$ is the closure of the set of fixed points on $S^2$ of elements of infinite order; equivalently, the accumulation set of any $\Gamma$-orbit. When $\Lambda\ne S^2$ the manifold has infinite volume, and $\Gamma$ acts properly discontinuously on the domain of discontinuity $\Omega=S^2\setminus\Lambda$. The convex core $\mathrm{core}(M)\subset M$ is the quotient of the convex hull of $\Lambda$. The group $\Gamma$ is convex cocompact if $\mathrm{core}(M)$ is compact, geometrically finite if $\mathrm{core}(M)$ has finite volume (equivalently, $\Gamma$ has a finite-sided fundamental polyhedron), and degenerate (geometrically infinite) if $\Gamma$ is finitely generated but not geometrically finite.

A geodesic plane in $M$ is the image of a nonconstant totally geodesic immersion $f\colon\mathbb{H}^2\to M$. A lift of $f$ to $\mathbb{H}^3$ has image a totally geodesic plane $P\subset\mathbb{H}^3$ bounded by a round circle $C\subset S^2$. The stabilizer of the circle is $\Gamma^C=\{\gamma\in\Gamma:\gamma(C)=C\}$; it acts on $P\cong\mathbb{H}^2$ as a discrete group of isometries — a Fuchsian-type group — and $f$ factors through the intrinsic surface $S=\Gamma^C\backslash\mathbb{H}^2$ (for $\Gamma$ torsion-free, $\pi_1(S)\cong\Gamma^C$), which is immersed in $M$. A discrete group of isometries of $\mathbb{H}^2$ is finitely generated if and only if its quotient surface has finite topological type (finite genus, finitely many ends); thus asking whether $\Gamma^C$ can be infinitely generated is asking whether an immersed geodesic plane can have infinite-type intrinsic topology even when the ambient group $\Gamma$ is finitely generated. The group $\Gamma^C$ is elementary if its limit set has at most two points (it is then virtually abelian) and nonelementary otherwise; its limit set always lies in $C\cap\Lambda$.

The space $\mathcal{C}$ of circles on $S^2$ identifies with the homogeneous space $G/H$ for $G=\mathrm{PGL}_2(\mathbb{C})$ and $H\cong\mathrm{PGL}_2(\mathbb{R})$ the stabilizer of a fixed circle, so $\mathcal{C}$ is also a space of frames; circle orbits $\Gamma\cdot C$ then correspond to $H$-invariant sets in the frame bundle $\mathrm{F}M=\Gamma\backslash G$, and the dynamics of $H$ (in particular its unipotent flows, meaning flows generated by one-parameter subgroups of unipotent, parabolic-type matrices) governs closures of planes. In finite-volume quotients, orbit rigidity for unipotent flows due to Shah and Ratner implies that a totally geodesic plane is either closed or dense, and a closed plane has finite area, hence a finitely generated (lattice) stabilizer. The infinite-volume setting is far subtler and was taken up by McMullen–Mohammadi–Oh: for $M$ rigid acylindrical — convex cocompact; acylindrical, meaning the compact 3-manifold bounded by the convex core carries no essential cylinders (embedded incompressible annuli not parallel into the boundary); and with $\Omega$ a disjoint union of round disks, so that $\Lambda$ is a round Sierpiński curve — they proved that every geodesic plane is closed or dense in the convex core, classified the possible orbit-closure types into five cases, and showed that exotic circles do not exist, where $C$ is exotic if $|C\cap\Lambda|>1$ and the orbit $\Gamma\cdot C$ contains no closed orbit. Their Theorem 6.2 shows that if $\Gamma$ is convex cocompact and $\Gamma\cdot C$ is discrete in $\mathcal{C}$, then $\Gamma^C$ is finitely generated and its limit set is exactly $C\cap\Lambda$: discreteness of the orbit makes the plane properly immersed, so a compact core gives finite generation of $\pi_1$. Within their classification, the stabilizer is pinned down in the discrete-orbit cases (where it is convex cocompact Fuchsian or elementary); the dense-orbit case leaves $\Gamma^C$ uncontrolled.

At the end of their introduction they ask: what can one say, in general, about the stabilizer $\Gamma^C\subset\Gamma$ of a circle — for example, can $\Gamma^C$ be infinitely generated? Two remarks calibrate the question. First, if $\Gamma$ itself is allowed to be infinitely generated, the answer is trivially yes: an infinite-type Fuchsian group preserves its own limit circle, which is contained in $\Lambda$. The substantive reading — the one intended, as the authors' cited analogy makes clear — takes $\Gamma$ finitely generated. Second, the analogy: McMullen proved that for the complexification $V_\varphi=\mathbb{H}/\Gamma_\varphi\to\mathcal{M}_2$ of a Teichmüller geodesic in the genus-2 moduli space $\mathcal{M}_2$ of Riemann surfaces (Teichmüller space, the universal cover of $\mathcal{M}_2$, parametrizes marked conformal structures on a fixed surface), the deck group $\Gamma_\varphi$ (conjugate to the affine group of the underlying form) has limit set the full circle $\partial\mathbb{H}$, yet is infinitely generated for infinitely many geodesics. So circle-type stabilizers with full-circle limit set genuinely can be infinitely generated one complexity level up, and there is no obvious reason hyperbolic 3-manifolds should be exempt.

Since 2017 the geodesic-plane program has advanced considerably — to all convex cocompact acylindrical manifolds, to geometrically finite manifolds with parabolics, to quantitative isolation of planes, to degenerate manifolds, and to higher-rank analogues — and several of these works establish finite generation of $\Gamma^C$ in their settings, always under a closed-orbit or in-the-convex-core hypothesis. The generation question in full generality is untouched.

Problem Statement

Let $\Gamma\subset\mathrm{PSL}_2(\mathbb{C})$ be a finitely generated Kleinian group acting on $\mathbb{H}^3$, let $C\subset S^2$ be a circle bounding a totally geodesic plane in $\mathbb{H}^3$, and let $\Gamma^C=\{\gamma\in\Gamma:\gamma(C)=C\}$ be its stabilizer, a discrete group of isometries of the plane. The problem, as posed by McMullen–Mohammadi–Oh, is to determine what can be said in general about the structure of $\Gamma^C$, and in particular:

(1) Can $\Gamma^C$ be infinitely generated? Either construct a finitely generated Kleinian group $\Gamma$ and a circle $C$ with $\Gamma^C$ infinitely generated, or prove that $\Gamma^C$ is finitely generated for every circle $C$ and every finitely generated $\Gamma$. The unrestricted version, with $\Gamma$ allowed to be infinitely generated, is trivially affirmative — an infinite-type Fuchsian group is its own circle stabilizer — so the finitely generated reading is the substantive one, consistent with the Teichmüller-space analogy drawn in the source.

(2) More broadly, characterize the possible structure of $\Gamma^C$: its limit set $\Lambda(\Gamma^C)\subset C\cap\Lambda(\Gamma)$ (in known cases it is $C\cap\Lambda$ itself, its non-isolated points, or a singleton), which parabolics it can contain, and which Fuchsian types (elementary, finite-area lattice, convex cocompact, degenerate, infinite type) can occur — especially for circles whose orbit $\Gamma\cdot C$ is not discrete in the space of circles and for degenerate (geometrically infinite) $\Gamma$, the two configurations in which nothing is currently known.

These linked subquestions form one objective: a structure theory for circle stabilizers in finitely generated Kleinian groups, with the highlighted subquestion (1) — infinite generation — as its concrete core and subquestion (2) as the broader structural question.

References

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