Free products from spinning and rotating families

Using geometry to understand the algebraic properties of a group is a primary aim of geometric group theory. This paper focuses on detecting when a group has the structure of a free product. The following theorem follows from Bass-Serre theory.

Dahmani–Guirardel–Osin [DGO17], inspired by the ideas of Gromov [Gro01], provided a far-reaching generalization of the theorem above. The simplicial tree above is replaced by a $\delta$-hyperbolic space, and the group acts via a very rotating family of subgroups. Under these conditions, they conclude that the group is a free product of subgroups in the family.

The set of apices $C\subset X$ (and also the pair $\mathcal{C}$) is $\rho$-separated if $d(c,c^{\prime})\geq\rho$ for all distinct $c,c^{\prime}\in C$. The family $\{G_{c}\}_{c\in C}$ of subgroups of $G$ is rotating if

1. (i)

   $C$ is $G$-invariant,

2. (ii)

   $G_{c}$ fixes $c$, for every $c\in C$,

3. (iii)

   $G_{g(c)}=gG_{c}g^{-1}$ for every $g\in G$ and $c\in C$.

Note that $G_{c}$ is a normal subgroup of the stabilizer $Stab_{G}(c)$ and similarly the subgroup ${\langle}G_{c}\mid c\in C{\rangle}$ of $G$ generated by all of the $G_{c}$ is normal in $G$. A rotating family is very rotating if in addition

1. (iv)

   for any distinct $c,c^{\prime}\in C$ and every $g\in G_{c}\setminus\{1\}$ every geodesic between $c^{\prime}$ and $g(c^{\prime})$ passes through $c$.

Note that (iv) is a bit weaker in the presence of sufficient separation than the definition in [DGO17]; see [DGO17, Lemma 5.5]. As an application, Dahmani–Guirardel–Osin solve a long-standing open problem by showing that the normal closure of a suitable power of any pseudo-Anosov mapping class in a mapping class group is free and all nontrivial elements in the normal closure are pseudo-Anosov. We discuss this in more detail below.

An important variation of the Dahmani–Guirardel–Osin theorem was recently proved by Clay–Mangahas–Margalit [CMM21]. In that setting, the group $G$ acts on a projection complex via a spinning family of subgroups. As an application, they determine the isomorphism type of the normal closure of a suitable power of various kinds of elements in the mapping class group. We will discuss this in more detail below as well. See related work in [Dah18, DHS20, CM].

We next explain the terminology in the above theorem. The projection data is a collection of metric spaces $\mathcal{Y}=\{X,Y,Z,\cdots\}$ (with infinite distance within a metric space allowed) together with “projections” $\pi_{X}(Y)\subset X$ for $X,Y\in\mathcal{Y}$ distinct, satisfying the following projection axioms for some $\theta\geq 0$ (called the projection constant, where we set $d_{X}(Y,Z)=\operatorname{diam}(\pi_{X}(Y)\cup\pi_{X}(Z))$:

1. (P1)

   $\operatorname{diam}\pi_{X}(Y)\leq\theta$, for any $X\neq Y$,

2. (P2)

   (the Behrstock inequality) if $d_{X}(Y,Z)>\theta$ then $d_{Y}(X,Z)\leq\theta$, and

3. (P3)

   for any $X,Y$ the set $\{Z\neq X,Y\mid d_{Z}(X,Y)>\theta\}$ is finite.

From this data, Bestvina–Bromberg–Fujiwara [BBF15] construct a graph $\mathcal{P}=\mathcal{P}(\mathcal{Y})$, called the projection complex, with the vertices in 1-1 correspondence with the spaces in $\mathcal{Y}$. Roughly speaking, $X$ and $Y$ are connected by an edge if $d_{Z}(X,Y)$ is small for any $Z$. This graph is connected, and it is quasi-isometric to a tree. Any group $G$ acting by isometries on the disjoint union $\bigsqcup_{X\in\mathcal{Y}}X$, permuting the spaces and commuting with projections (i.e. $g(\pi_{X}(Y))=\pi_{g(X)}g(Y)$), acts by isometries on $\mathcal{P}$, and we say that $G$ preserves the projection data.

An $L$-spinning family is a family $\{G_{c}\}$ parametrized by the vertices $c\in V\mathcal{P}$ satisfying

1. (a)

   $G_{c}$ fixes $c$,

2. (b)

   $G_{g(c)}=gG_{c}g^{-1}$ for $g\in G$, $c\in C$, and

3. (c)

   $d_{c}(c^{\prime},g(c^{\prime}))>L$ for $c\neq c^{\prime}$ and $g\in G_{c}\smallsetminus\{1\}$.

The main goal of this paper is to simplify and significantly shorten the proof of the Dahmani–Guirardel–Osin theorem using the Clay–Mangahas–Margalit theorem and the machinery of projection complexes. We also present a variant of the proof of the [CMM21] theorem to directly construct an action of the group on a tree as in Theorem 1.1. Given a group action on a $\delta$-hyperbolic metric space equipped with a very rotating family of subgroups, we construct an action of that group on a projection complex with the same family acting as a spinning family. While our proof of Theorem 1.3 still uses the construction of windmills (which are used in [DGO17] and [Gro01]), our work differs from [CMM21] in that we find a natural tree on which $G$ acts as in Theorem 1.1 and eliminate the need to work with normal forms. We also introduce the notion of canoeing in a projection complex, which is inspired by the classic notion of canoeing in the hyperbolic plane (see Section 4), and enables us to further streamline some of the arguments from [CMM21].

To prove Theorem 1.4, we construct a projection complex via the Bestvina–Bromberg–Fujiwara axioms. These axioms require us to first define for each $c\in C$ a metric space $S_{c}$ (with infinite distances allowed) and projections $\pi_{S_{c}}(S_{c^{\prime}})$, which we abbreviate to $\pi_{c}(c^{\prime})$. A standard example of such a construction is the following. Take a closed hyperbolic surface $S$ and a closed (not necessarily simple) geodesic $\alpha$. Consider the universal cover $\tilde{S}=\mathbb{H}^{2}$ and the set $\mathcal{Y}$ of all lifts of $\alpha$. For two different lifts $A,B\in\mathcal{Y}$, define $\pi_{A}(B)\subset A$ to be the nearest point projection of $B$ to $A$. This will be an open interval in $A$ whose diameter is uniformly bounded independently of $A,B$ (but which depends on $\alpha$). Roughly speaking, $\pi_{A}(B)$ can have a large diameter only of $B$ fellow travels $A$ for a long time. It is not hard to see that the projection axioms hold in this case. A similar construction can be carried out when $S$ is a hyperbolic surface with a cusp and $\alpha$ is a horocycle. Now $\mathcal{Y}$ is an orbit of pairwise disjoint horocycles in $\mathbb{H}^{2}$ and $\pi_{A}(B)$ is defined as the nearest point projection of $B$ to $A$ as before. There are now two natural choices of a metric on horocycles in $\mathcal{Y}$: one can take the intrinsic metric so that it is isometric to $\mathbb{R}$ or the induced metric from $\mathbb{H}^{2}$. Either choice satisfies the projection axioms, but note here that the intrinsic metric can also be defined as the path metric where paths are not allowed to intersect the open horoball cut out by the horocycle.

The starting point of our proof of Theorem 1.2 is the construction of the projection complex whose vertex set is the set $C$ of apices, inspired by the horocycle example. To each $c\in C$ we associate the sphere $S_{c}$ of radius $R$ centered at $c$. The number $R$ is chosen carefully. The open balls $B_{c}$ cut out by the spheres should be pairwise disjoint and a reasonable distance apart (a fixed multiple of $\delta$), yet big enough so that paths in the complement of $B_{c}$ joining points $x,y\in S_{c}$ on opposite sides of the ball are much longer (exponential in $R$) than a geodesic in $X$ joining $x,y$ (which is linear in $R$). The projection $\pi_{c}(c^{\prime})$ for $c,c^{\prime}\in\mathcal{C}$, $c\neq c^{\prime}$ is the set of all points in $S_{c}$ that lie on a geodesic between $c$ and $c^{\prime}$. The metric we take on $S_{c}$ is induced by the path metric in $X\smallsetminus B_{c}$ (this can take value $\infty$ if the ball disconnects $X$). We check that with these definitions the projection axioms hold (see Section 3.1). Thus, the group $G$ acts on the projection complex and we check that the groups $G_{c}$ form a spinning family (see Section 3.2), which proves Theorem 1.4.

The same proof goes through with a slightly weaker hypothesis that the family $\{G_{c}\}$ is fairly rotating, instead of very rotating (with slightly different constants). Here we require only that geodesics between $c^{\prime}$ and $g(c^{\prime})$ pass within $1$ of $c$, for $g\in G_{c}\setminus\{1\}$, instead of passing through $c$. This situation naturally occurs. As an example, consider a closed hyperbolic orbifold $S$ with one cone point, with cone angle $2\pi/n$ for $n>2$. The orbifold universal cover $\tilde{S}=\mathbb{H}^{2}$ admits an action by the orbifold fundamental group $G$ where the stabilizers $G_{c}\cong\mathbb{Z}/n\mathbb{Z}$ of the lifts $c\in C$ of the cone point form a rotating family. This family will never be very rotating, but it will be fairly rotating if the pairwise distance between distinct elements of $C$ is large enough, given by a function of $n$.

Note that in Theorem 1.3, the constant $L(\mathcal{P})$ really depends only on the projection constant $\theta$ and can be taken to be a fixed multiple of $\theta$ (e.g. $1000\theta$ will do). In Theorem 1.4 the projection constant $\theta$ in (1) can be taken to be a fixed multiple of $\delta$, and $L(\rho)$ in (2) will be an exponential function in $\rho$. Since exponential functions grow faster than linear functions, the spinning constant $L$ in Theorem 1.4 will beat the one in Theorem 1.3 if $\rho$ is big enough, so Theorem 1.2 will follow (see Section 5).

We now say a few words about our proof of Theorem 1.3. As in [CMM21], we recursively define a sequence of windmills which correspond to certain orbits of larger and larger collections of the vertex subgroups $\{G_{c}\}$. At each stage we prove that the these windmills have a tree-like structure (technically, the skeleton of the canonical cover of each windmill is a tree). At each step we obtain a new group that is the free product of the previous group with a suitable collection of $G_{c}$’s, and taking the limit proves the theorem, see Section 4.2. Canoeing enters when we verify that windmills have a tree-like structure. The simplest example of a canoeing path in $\mathcal{P}$ would be an edge-path passing through vertices $v_{1},v_{2},\cdots,v_{k}$ such that for every $i=2,3,\cdots,k-1$ the “angle” $d_{v_{i}}(v_{i-1},v_{i+1})$ is large. The basic properties of projection complexes quickly imply that such paths are embedded, and they provide a local-to-global principle enabling us to establish the tree-like structure.

We end this introduction with some applications of Theorems 1.2 and 1.3. Suppose $G$ acts by isometries on a hyperbolic space $Y$ and $g\in G$ is loxodromic. Suppose also that $g$ is a “WPD element” as per Bestvina-Fujiwara [BF02]. This amounts to saying that $g$ is contained in a unique maximal virtually cyclic subgroup $EC(g)$ (the elementary closure of $\langle g\rangle$) and further that the set of $G$-translates of a fixed $EC(g)$-orbit is “geometrically separated”, i.e., the nearest point projections satisfy the projection axioms. This situation generalizes the example above, where $Y=\mathbb{H}^{2}$, $G$ is the deck group of the universal cover $Y\to S$, and $g$ corresponds to an indivisible element in $G$, so $EC(g)=\langle g\rangle$. This situation is fairly common. For example, $Y$ could be the curve complex of a surface of finite type, $G$ its mapping class group, and $g\in G$ a pseudo-Anosov mapping class (see [BF02]). For another example, take $G$ to be the Cremona group (of birational transformations of $\mathbb{C}\mathbb{P}^{2}$) acting on infinite dimensional real hyperbolic space, see [CLC13]. In these situations one can construct a space $X$ by coning off the orbit of $EC(g)$ and each translate. If the radius of the cone is large enough, Dahmani–Guirardel–Osin show that $X$ is hyperbolic and if $N\unlhd EC(g)$ is a sufficiently deep finite index normal subgroup, then the set of $G$-conjugates of $N$ forms a very rotating family with the cone points as apices. In particular, they resolved a long-standing open problem by showing that if $g$ is a pseudo-Anosov homeomorphism of a finite-type surface, then there is $n>0$ such that the normal closure of $g^{n}$ in the mapping class group is the free group $F_{\infty}$ (of infinite rank), and all nontrivial elements of the group are pseudo-Anosov.

Clay–Mangahas–Margalit reproved this application to mapping class groups directly from Theorem 1.3 and gave new applications of their theorem. To illustrate, consider a mapping class $g$ on a finite-type surface $S$ which is supported on a proper, connected, $\pi_{1}$-injective subsurface $A\subset S$ such that $g|A$ is a pseudo-Anosov homeomorphism of $A$. Assume also that any two subsurfaces in the orbit of $A$ either coincide or intersect. There is a natural projection complex one can construct from this setup. The vertices are the subsurfaces in the mapping class group orbit of $A$, and the projection $\pi_{A}(B)$ is the Masur-Minsky subsurface projection [MM99, MM00]. It follows from the work of Masur-Minsky and Behrstock that the projection axioms hold in this setting. Clay–Mangahas–Margalit prove that for a suitable $n>0$ the collection of conjugates of $\langle g^{n}\rangle$ forms a spinning family and conclude that, here too, the normal closure of $g^{n}$ is free. They consider more general situations where the normal closure can be a non-free group as well. Remarkably they can exactly determine the normal closure even in this case. For example, if $S$ is a closed surface of even genus and $g$ is pseudo-Anosov supported on exatly half the surface, then for suitable $n>0$ the normal closure of $g^{n}$ is the infinite free product of copies of $F_{\infty}\times F_{\infty}$.

In this section, we state the relevant result of Dahmani–Guirardel–Osin, give background on projection complexes, state the result of Clay–Mangahas–Margalit, and give the necessary background on $\delta$-hyperbolic spaces, in that order.

In this section we construct a projection complex from a fairly rotating family. Throughout, let $G$ be a group that acts by isometries on a $\delta$-hyperbolic metric space $X$. Let $\mathcal{C}=(C,\{G_{c}\,|\,c\in C\})$ be a $\rho$-separated fairly rotating family for some $\rho\geq 20\delta$.

We think of the associated projection distances, $d_{p}(b,c)=\operatorname{diam}(\pi_{p}(b)\cup\pi_{p}(c))$, as the penalty (up to an error of a fixed multiple of $\delta$) of traveling from $b$ to $c$ avoiding a ball of fixed radius around $p$.

The aim of this section is to prove the following theorem.

We prove the projection axioms are satisfied in Subsection 3.1, and we verify the spinning condition in Subsection 3.2.