Free Groups as End Homogeneity Groups of $3$-manifolds

We refer to [12, 13, 8, 11] for a discussion of Cantor sets in general and of rigid Cantor sets, and to [24] for results about local genus of points in Cantor sets and defining sequences for Cantor sets. The bibliographies in these papers contain additional references to results about Cantor sets. Two Cantor sets $X$ and $Y$ in $\mathbb{R}^{3}$ (or $S^{3}$) are said to be equivalent or equivalently embedded if there is a self-homeomorphism of $\mathbb{R}^{3}$ (or $S^{3}$) taking $X$ to $Y$. If there is no such homeomorphism, the Cantor sets are said to be inequivalent, or inequivalently embedded. A Cantor set $C$ is said to be rigidly embedded in $\mathbb{R}^{3}$ (or $S^{3}$) if the only self-homeomorphism of $C$ that extends to a homeomorphism of $\mathbb{R}^{3}$ (or $S^{3}$) is the identity.

We list the results which we will need on geometric index. See Schubert [17, 8] for more details. If $K$ is a link in the interior of a solid torus $T$, the geometric index of $K$ in $T$, denoted by $\text{N}(K,T)$, is the minimum of $|K\cap D|$ over all meridional disks $D$ of $T$ intersecting $K$ transversely. If $T$ is a solid torus and $M$ is a finite union of disjoint solid tori so that $M\subset\text{Int}\ T$, then the geometric index $\text{N}(M,T)$ of $M$ in $T$ is $\text{N}(K,T)$ where $K$ is a core of $M$.

Note: An unknotted solid torus is a solid torus $B^{2}\times S^{1}$ with center line $\{0\}\times S^{1}$ an unknotted circle in $S^{3}$ .

There is one additional result we will need.

We review the definition and some basic facts from [24] about the local genus of points in a Cantor set. Also see [24] for a discussion of defining sequences.

A defining sequence for a Cantor set $X$ in $\mathbb{R}^{3}$ is a properly nested sequence of handlebodies $(M_{i})$ such that $\bigcap M_{i}=X$. Let ${\mathcal{D}}(X)$ be the set of all defining sequences for a Cantor set $X\subset S^{3}$. Let $(M_{i})\in{\mathcal{D}}(X)$ be one of these defining sequences. For $A\subset X$, denote by $M_{i}^{A}$ the union of those components of $M_{i}$ which intersect $A$. Define

where $g(M_{i}^{A})$ is the genus of $M_{i}^{A}$. The number $g_{A}(X)$ is called the genus of the Cantor set $X$ with respect to the subset $A$. For $A=\{x\}$ we call the number $g_{\{x\}}(X)$ the local genus of the Cantor set $X$ at the point $x$ and denote it by $g_{x}(X)$.

The following result from [24] is needed to show that certain points in our examples have local genus $>1$.

For background on Freudenthal compactifications and theory of ends, see [4, 7, 20]. For an alternate proof using defining sequences of the result that every homeomorphism of the open $3$-manifold extends to a homeomorphism of its Freudenthal compactification, see [9].

Each Cantor set $C$ in $S^{3}$ has for its complement an open $3$-manifold $M^{3}$ with end set $C$. The embedding of the Cantor set gives rise to properties of the corresponding complementary $3$-manifold $M^{3}$. See [9, 11, 21] for examples of this.

We investigate possible group actions on the end set $C$ of the open $3$-manifold $M^{3}$ in the following sense: The homogeneity group of the end set is the group of homeomorphisms of the end set $C$ that extend to homeomorphisms of the open $3$-manifold $M^{3}$. Referring specifically to the embedding of the Cantor set, this group can also be called the embedding homogeneity group of the Cantor set. See [5, 22] for a discussion and overview of some other types of homogeneity.

The standardly embedded Cantor set is at one extreme here. The embedding homogeneity group is the full group of self-homeomorphisms of the Cantor set, an extremely rich group (there is such a homeomorphism taking any countable dense subset to any other). Cantor sets with this full embedding homogeneity group are called strongly homogeneously embedded. See Daverman [3] for an example of a non-standard Cantor set with this property.

At the other extreme are rigidly embedded Cantor sets, i.e. those Cantor sets for which only the identity homeomorphism extends. Shilepsky [19] constructed Antoine type [2] rigid Cantor sets. Their rigidity is a consequence of Sher’s result [18] that if two Antoine Cantor sets are equivalently embedded, then the stages of their defining sequences must match up exactly. Newer examples [13, 8] of non-standard Cantor sets were constructed that are both rigidly embedded and have simply connected complement. See [23] for additional examples of rigidity.

These examples naturally lead to the question of which types of groups can arise as end homogeneity groups between the two extremes mentioned above. In this paper we show that for each finitely generated free group $G$, there is an irreducible open 3-manifold with the end set homeomorphic to a Cantor set and with the end homogeneity group isomorphic to $G$.

An Antoine Cantor set is described as follows. Let $M_{0}$ be an unknotted solid torus in $S^{3}$. Let $M_{1}$ be a chain of at least four linked, pairwise disjoint, unknotted solid tori in $M_{0}$ as in Figure 1. Inductively, $M_{i}$ consists of pairwise disjoint solid tori in $S^{3}$ and $M_{i+1}$ is obtained from $M_{i}$ by placing a chain of at least four linked, pairwise disjoint, unknotted solid tori in each component of $M_{i}$. If the diameter of the components goes to $0$, the Antoine Cantor set is $C=\bigcap\limits_{i=0}^{\infty}M_{i}$.

We refer to Sher’s paper [18] for basic results and description of Antoine Cantor sets. The key result we will need is the following:

In particular, the number and adjacency of links in the chains must match up at each stage.

We inductively describe how to label the vertices and how to label and orient the edges of the graph described above.

Begin by labeling the vertex at the origin corresponding to the identity as $v(e)$. Label the vertex at $p(1,j)$ for $1\leq j\leq 2N$ as $v(\beta_{j})$.

Label the edge from $v(e)$ to $v(\beta_{j})$, $1\leq j\leq 2N$ with the generator $\beta_{j}$ and orient it from $v(e)$ to $v(\beta_{j})$. Note that this is equivalent to labeling the edge from $v(\beta_{j})$ to $v(e)$ with the generator $\overline{\mathstrut\beta_{j}}$ and orienting it from $v(\beta_{j})$ to $v(e)$. For notation, we refer to this edge as $E(v(e),\beta_{j})$ or $E(v(\beta_{j}),\overline{\mathstrut\beta_{j}})$ to emphasize the initial vertex and generator needed to get to the other vertex. To emphasize the initial and final vertices, we can use the notation $E(v(e),v(\beta_{j}))=E(v(\beta_{j}),v(e))$.

Note that starting at any point on the circle of radius $1/2$ centered at the origin and traveling in the counterclockwise direction around this circle, one encounters the labeled edges in the cyclic order $E(v(e),\beta_{1}),\ldots,E(v(e),\beta_{2N})$, or equivalently, by Remark 2.1, in the cyclic order:

$E(v(e),\beta_{1}),\ldots,E(v(e),\beta_{N}),E(v(\beta_{N+1}),\beta_{1}),\ldots,E(v(\beta_{2N}),\beta_{N})$.

Inductively assume the vertices at distance $m$ from the origin, $m\leq n$, have been labeled as $v(g_{i})$, where $g_{i}$ is a reduced word of length $m$. Also assume that edges from vertices $v(g_{i})$ at distance $m$ from the origin $(m\leq n-1)$ to $v(g_{\ell})$ at distance $m+1$ from the origin have been labeled $E(v(g_{i}),\beta_{k})$ where $g_{\ell}=g_{i}\circ\beta_{k}$.

Consider $v(g)$ for some $g$ of reduced length n. Then $g=g^{\,\prime}\circ\beta_{k}$ for some reduced word $g^{\,\prime}$ of length $n-1$ and some $\beta_{k}\in g(F_{N})$. The edge joining $v(g^{\,\prime})$ to $v(g)$ is labeled $E(v(g^{\,\prime}),\beta_{k})$, or equivalently $E(v(g),\overline{\mathstrut\beta_{k}})=E(v(g),\beta_{k+N\text{(mod 2N)}})$. Let $j=k+n\text{(mod 2N)}$.

Form a circle of radius $1/2$ about $v(g)$. Starting at the edge $E(v(g),\beta_{j})$, proceed around the circle in the counterclockwise direction labeling the edges encountered as

all subscripts mod(2N).

Label the vertices at distance $n+1$ from the origin as follows. If $v$ is such a vertex, then there is an edge $E(v(g),\beta_{j})$ from a vertex $v(g)$ at distance $n$ from the origin to $v$. Label the vertex $v$ as $v(g\circ\beta_{j})$.

The inductive assumption is easily checked for the newly labeled vertices and edges. See Figure 3 for an illustration, keeping in mind Remark 2.1.

For each element $g$ of $F_{N}$ we define a graph automorphism $h_{g}$ of $G(F_{N})$ as follows. For a vertex $v({g_{i}})$ of $G(F_{N})$ corresponding to the element $g_{i}$ of $F_{N}$, $h_{g}(v({g_{i}}))$ is defined to be $v(g\circ g_{i})$. The edge from $v(g_{i})$ to $v(g_{i}\circ\beta_{k})$ can be labeled $E(v(g_{i}),\beta_{k})$ when oriented from $v(g_{i})$ to $v(g_{i}\circ\beta_{k})$. Define $h_{g}(E(v(g_{i}),\beta_{k})$ to be the linear homeomorphism from $E(v(g_{i}),\beta_{k})$ to $E(v(g\circ g_{i}),\beta_{k})$.

The graph automorphisms $h_{\beta_{i}}$, for $1\leq i\leq N$, can be extended to homeomorphisms ${h^{\prime}}_{\beta_{i}}$ from $\mathbb{R}^{2}$ to itself. Note that $\mathbb{R}^{2}-G(F_{N})$ consists of a collections of unbounded regions with boundary in $G(F_{N})$. Adding in the point at infinity, each region is a closed disc with boundary consisting of edges in $G(F_{N})$ together with this added point. The graph automorphism $h_{\beta_{i}}$ takes the boundary of one of these disc regions to the boundary of another, and can thus be extended to the entire region. There are many choices for these extension homeomorphisms. They coincide up to proper isotopy.

Piecing together these extensions, one gets an extension of $h_{\beta_{i}}$ to a self homeomorphism of $\mathbb{R}^{2}$, ${h^{\prime}}_{\beta_{i}}$ for $1\leq i\leq N$. Define ${h^{\prime}}_{\beta_{i}}$ for $N+1\leq i\leq 2N$ to be $(h^{\prime}_{\beta_{i-N}})^{-1}$. Since $h_{\beta_{i}}=(h_{\beta_{i-N}})^{-1}$ for $N+1\leq i\leq 2N$ as graph automorphisms, it follows that $h^{\prime}_{\beta_{i}}$ is an extension of $h_{\beta_{i}}$ for $N+1\leq i\leq 2N$.

For a general $g\in F_{N}$, $g$ has a unique reduced representation $g=\beta_{i1}\circ\beta_{i2}\circ\ldots\circ\beta_{iM}$ where the $\beta_{ij}$ are in $g(F_{N})$. Define $h^{\prime}(g)$ to be:

The homeomorphism $h^{\prime}_{g}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ can then be extended to a homeomorphism $H_{g}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ by crossing with the identity.

In $\mathbb{R}^{3}$, choose a 3-ball $B(v(e))$ around the origin $=v(e)$ so that the homeomorphic copies $H_{\beta_{i}}(B(v(e)))$, $1\leq i\leq 2N,$ together with $B(v(e))$ are all pairwise disjoint. Let $B(v(\beta_{i}))$ denote the 3-ball $H_{\beta_{i}}(B(v(e)))$. The original ball $B(v(e))$ can be chosen so that it is contained in a ball of radius $1/5$ about $v(e)$, and so that each $B(v(\beta_{i}))$ is contained in a ball of radius $1/5$ about $v(\beta_{i})$.

For a general $g\in F_{N}$, use the unique reduced representation of $g$ as $\beta_{i1}\circ\beta_{i2}\circ\ldots\circ\beta_{iM}$ where the $\beta_{ij}$ are in $g(F_{N})$. The ball $B(v(g))$ is then defined to be to be:

We similarly define a collection of tubes. For each $i$, $1\leq i\leq N$ choose a thin tube $D(v(e),\beta_{i})$ joining $B(v(e))$ to $B(v(\beta_{i}))$ so that $B(v(e))\cup D(v(e),\beta_{i})\cup B(v(\beta_{i}))$ contains $E(v(e),\beta_{i})$. Choose the tubes $D(v(e),\beta_{1}),\ldots,D(v(e),\beta_{N})$, so as to be pairwise disjoint. For each $i$, $N+1\leq i\leq 2N$ let $D(v(e),\beta_{i})$ be $H_{\beta_{i}}(D(v(e),\beta_{i-N}))=H_{\beta_{i}}(D(v(e),\overline{\beta_{i}}))$. See Figure 4 for an illustration for $F_{2}$.

We will now define rigid Antoine Cantor sets associated with each vertex in $G(F_{N})$, and a chain of three linked rigid Cantor sets for each edge in $G(F_{n})$. See Figure 5 for an illustration of the outer tori in defining sequences for these rigid Cantor sets in the case of $F_{2}$.

To begin, construct a rigid Antoine Cantor set $C(v(e))$ so the outer torus, $T(C(v(e)))$, of the construction is in $B(v(e))$. This $T(C(v(e)))$ corresponds to the central torus in Figure 5 in the case of $F_{2}$. Let $T^{\prime}(C(v(e)))$ be the union of the second stage tori in the construction of $C(v(e))$. See Figure 1 for an illustration of the first and second stages of an Antoine construction.

For a general $g\in F_{N}$, use the unique reduced representation of $g$ as $\beta_{i1}\circ\beta_{i2}\circ\ldots\circ\beta_{iM}$ where the $\beta_{ij}$ are in $g(F_{N})$. Define a rigid Cantor set $C(v(g))\subset B(v(g))$ to be:

Similarly, let $T(C(v(g)))=H_{g}(T(C(v(e))))\text{ and }T^{\,\prime}(C(v(g)))=H_{g}(T^{\prime}(C(v(e)))).$ The four outer tori in Figure 5 correspond to $T(C(a_{1})),T(C(a_{2})),T(C(\overline{\mathstrut a_{1}}))$, and $TC(\overline{\mathstrut a_{2}})$ in the case of $F_{2}$.

We next construct rigid Cantor sets $C_{1}(v(e),\beta_{k}),C_{2}(v(e),\beta_{k}),C_{3}(v(e),\beta_{k})$ for each $k,1\leq k\leq N$. Choose a series of three linked tori, $T_{1}(v(e),\beta_{k}),T_{2}(v(e),\beta_{k}),$ and $T_{3}(v(e),\beta_{k})$ joining $T(C(v(e)))\text{ and }T(C(\beta_{k})),1\leq k\leq N$ and contained in $B(v(e))\cup D(v(e),\beta_{k})\cup B(v(\beta_{k}))$ as in Figure 5. Label the torus linked to $T(C(v(e)))$ as $T_{1}(v(e),\beta_{k})$, and use it as the outer torus for a rigid Antoine Cantor set $C_{1}(v(e),\beta_{k})$. Label the torus linked to $T(C(\beta_{k}))$ as $T_{3}(v(e),\beta_{k})$, and use it as the outer torus for a rigid Antoine Cantor set $C_{3}(v(e),\beta_{k})$. Label the remaining torus as $T_{2}(v(e),\beta_{k})$ and use it as the outer torus for a rigid Antoine Cantor set $C_{2}(v(e),\beta_{k})$.

For $N+1\leq k\leq 2N$, and $1\leq j\leq 3$, let $C_{j}(v(e),\beta_{k})=H_{\beta_{k}}(C_{j}(v(e),\overline{\beta_{k}}))$, and $T(C_{j}(v(e),\beta_{k}))=H_{\beta_{k}}(T(C_{j}(v(e),\overline{\beta_{k}})))$. This completes the definition of the $3\cdot 2N$ rigid Cantor sets along the edges $E(v(e),\beta_{k})$. See Figure 5 for outer tori corresponding to these Cantor sets in the case $F(2)$.

For $g\in F_{N}$, let $C_{i}(v(g),\beta_{k})=H_{g}(C_{i}(v(e),\beta_{k}))$ and $T(C_{i}(v(g),\beta_{k}))=H_{g}(T(C_{i}(v(e),\beta_{k})))$ for $1\leq i\leq 3$ and $1\leq k\leq 2N$.

For any rigid Antoine Cantor set $C$ in this construction, $T(C)$ represents the outer torus in the Antoine Construction and $T^{\prime}(C)$ represents the union of the second stage tori in the construction.

The union of the Cantor sets $C_{i}(g,\beta_{k})$ together with the union of the Cantor sets $C(v(g))$ is not a Cantor set since it is not compact. To remedy this, we add the point at infinity, denoted by $p$, to $\mathbb{R}^{3}$ to get $S^{3}$. We then define the Cantor set $\mathcal{C}$ associated with $G(F_{N})$ to be: $\mathcal{C}=C(G(F_{N}))=\{C_{i}(v(g),\beta_{k}),g\in F_{N},1\leq k\leq 2N\}\cup\{C(v(g)),g\in F_{N}\}\cup\{p\}.$