A Universal Cannon-Thurston map and the surviving curve complex.

Existence of the Cannon-Thurston map in the context of Kleinian groups is proved by several authors starting with Floyd [Flo80] for geometrically finite Kleinian groups and then by Cannon and Thurston for fibers of closed hyperbolic 3-manifolds fibering over the circle. Cannon and Thurston's work was circulated as a preprint around 1984 and inspired works of many others before it was published in 2007 [CT07]. The existence of the Cannon-Thurston map was proven by Minsky [Min94] for closed surface groups of bounded geometry and by by Mitra and Klarreich [Mit98b, Kla99a] for hyperbolic 3-manifolds of bounded geometry with an incompressible core and without parabolics. Alperin-Dicks-Porti [ADP99] proved the existence of the Cannon-Thurston map for figure eight knot complement, McMullen [McM01] for punctured torus groups, and then Bowditch [Bow07, Bow13] for more general punctured surface groups of bounded geometry. Mj completed the investigation for all finitely generated Kleinian surface groups without accidental parabolics, first for closed and then for punctured surfaces in a series of papers that culminated in the two papers [Mj14a] and [Mj14b], the latter with an appendix by S. Das. For general Kleinian groups, see Das-Mj [DM16] and Mj [Mj17], and the survey [Mj18].

Moving beyond real hyperbolic spaces, it is now classical that a quasi-isometric embedding of one Gromov hyperbolic space into another extends to an embedding of the Gromov boundaries. One of the first important generalizations of Cannon and Thurston's work outside the setting of Kleinian groups is due to Mitra in [Mit98a] who proved that given a short exact sequence

of infinite word hyperbolic groups, the Cannon–Thurston map exists and it is surjective. In this case the Cannon-Thurston map $\partial H\rightarrow\partial\Gamma$ is defined between the Gromov boundary $\partial H$ of the fiber group $H$ and the Gromov boundary $\partial\Gamma$ of its extension $\Gamma$. Mitra defined an algebraic ending lamination associated to points in the Gromov boundary of the base group $G$ in [Mit97], and recent work of Field [Fie20] proves that the quotient of $\partial H$ in terms of such an ending lamination is a dendrite (compare the Kleinian discussion above).

In a different direction, Mitra later extended his existence result to trees of hyperbolic spaces; see [Mit98b]. In 2013 Baker and Riley gave the first example example of a hyperbolic subgroup of a hyperbolic group with no continuous Cannon-Thurston map ([BR13]); see also Matsuda [MO14]. On the other hand, Baker and Riley ([BR20]) proved existence of Cannon-Thurston maps even under arbitrarily heavy distortion of a free subgroup of a hyperbolic group.

For free groups and their hyperbolic extensions, Cannon-Thurston maps are better understood than arbitrary hyperbolic extensions. Kapovich and Lustig characterized the Cannon-Thurston maps for hyperbolic free-by-cyclic groups with fully irreducible monodromy [KL15]. Later Dowdall, Kapovich and Taylor characterized Cannon-Thurston maps for hyperbolic extensions of free groups coming from convex cocompact subgroups of outer automorphism group of the free group [DKT16].

Finally we note that we have only discussed a few of the many results on the existence or structure of Cannon-Thurston maps in various settings. For more see e.g. [Mj14b, MR18, MP11, JKLO16, Gué16, Fen92, Fra15, Fen16, Mou18]).

In Section 2, we give preliminaries on curve complexes, witnesses and Gromov boundary of a hyperbolic space along with basics of spaces of laminations. In particular, subsection 2.4 is devoted to the construction of the survival map and in subsection 2.5 the relation between cusps and witnesses via the survival map is given. In Section 3, we define survival paths in ${\mathcal{C}^{s}(\dot{S})}$ and give an upper bound on the survival distance $d^{s}$ in terms of projection distances into curve complexes of witnesses. In Section 4 we prove the hyperbolicity of ${\mathcal{C}^{s}(\dot{S})}$. Section 5 is devoted to the distance formula for ${\mathcal{C}^{s}(\dot{S})}$, a-la Masur-Minsky, and as a result we prove that survival paths are uniform quasi-geodesics in ${\mathcal{C}^{s}(\dot{S})}$. In Section 6 we explore the boundary of the survival curve complex ${\mathcal{C}^{s}(\dot{S})}$ and prove that it is homeomorphic to the space of survival ending laminations on $\dot{S}$, a result analogous to that of Klarreich [Kla99b]. In Section 7 we extend the definition of survival map to the closures of curve complexes. Finally in Section 8, we prove Theorem 1.3 and the rest of the theorems from the introduction. Specifically, we prove the existence and continuity of the map $\partial\Phi$ in Section 8.1 and its surjectivity in Section 8.2. Finally, we Section 8.3 we prove the universal property of $\partial\Phi$ as well as constructing the map $\partial\Phi_{0}$.

The authors would like to thank Saul Schleimer for helpful conversations in the early stages of this work. The second author would also like to thank Autumn Kent, Mahan Mj, and Saul Schleimer for their earlier collaborations that served as partial impetus for this work.

Let $x,y,C,K\geq 0$ with $K\geq 1$. We write $x\stackrel{{\scriptstyle K,C}}{{\preceq}}y$ to mean $x\leq Ky+C$. We also write

When the constants are clear from the context or independent of any varying quantities and unimportant, we also write $x\preceq y$ as well as $x\asymp y$. In addition, we will use the shorthand notation ${\{\{}x{\}\}}_{C}$ denote the cut-off function giving value $x$ if $x\geq C$ and $0$ otherwise.

Any connected simplicial complex will be endowed with a path metric obtained by declaring each simplex to be a regular Euclidean simplex with side lengths equal to $1$. The vertices of a connected simplicial complex will be denoted with a subscript $0$, and the distance between vertices will be an integer computed as the minimal length of a path in the $1$–skeleton. By a geodesic between a pair of vertices $v,w$ in a simplicial complex, we mean either an isometric embedding of an interval into the $1$–skeleton with endpoints $v$ and $w$ or the vertices encountered along such an isometric embedding, with the situation dictating the intended meaning.

By a curve on a surface $Y$, we mean an essential (homotopically nontrivial and nonperipheral), simple closed curve. We often confuse a curve with its isotopy class. When convenient, we take the geodesic representative with respect to a complete finite area hyperbolic metric on the surface with geodesic boundary components (if any) and realize an isotopy class by its unique geodesic representative. A multi-curve is a disjoint union of pairwise non-isotopic curves, which we also confuse with its isotopy class and geodesic representative when convenient.

The curve complex of a surface $Y$ with $2\leq\xi(Y)<\infty$ is the complex ${\mathcal{C}}(Y)$ whose vertices are curves (up to isotopy) and whose $k$–simplices are multi-curves with $k+1$ components. According to work of Masur-Minsky [MM99], curve complexes are Gromov hyperbolic. For other proofs, see [Bow06, Ham07] as well as [Aou13, Bow14, CRS14, HPW15] which prove uniform bounds on $\delta$.

The surviving complex ${\mathcal{C}}^{s}(\dot{S})$ is defined to be the subcomplex of the curve complex ${\mathcal{C}}(\dot{S})$, spanned by those curves that do not bound a twice-punctured disk, where one of the punctures is the $z$–puncture. Given curves $\alpha,\beta\in{\mathcal{C}}^{s}_{0}(\dot{S})$, we write $d^{s}(\alpha,\beta)$ for the distance between $\alpha$ and $\beta$ (in the $1$–skeleton).

A subsurface of $\dot{S}$ is either $\dot{S}$ itself or a component $Y\subset\dot{S}$ of the complement of a small, open, regular neighborhood of a (representative of a) multi-curve $A$; we assume $Y$ is not a pair of pants (a sphere with three boundary components/punctures). The boundary of $Y$, denoted $\partial Y$, is the sub-multi-curve of $A$ consisting of those components that are isotopic into $Y$. As with (multi-)curves, subsurfaces is considered up to isotopy, in general, but when convenient we will choose a representative of the isotopy class without comment.

Clearly, $\dot{S}$ is a witness. Note that if $\beta$ is the boundary of a twice-punctured disk $D$, one of which is the $z$–puncture, and the complementary component $W\subset S$ with $\partial W=\beta$ is a witness. To see this, we observe that any curve $\alpha$ in ${\mathcal{C}^{s}(\dot{S})}$ that can be isotoped disjoint from $W$ must be contained in $D$, but the only such curve in $\dot{S}$ is $\beta$. It is clear that these two types of subsurfaces account for all witnesses. We let $\Omega(\dot{S})$ denote the set of witnesses and $\Omega_{0}(\dot{S})=\Omega(\dot{S})\smallsetminus\{\dot{S}\}$ the set of proper witnesses. We note that any proper witness $W$ is determined by its boundary curve, $\partial W$: if $W\neq\dot{S}$, then $W$ is the closure of the component of $\dot{S}{\smallsetminus}\partial W$ not containing the $z$–puncture.

An important tool in what follows is the subsurface projection of curves in ${\mathcal{C}^{s}(\dot{S})}$ to witnesses; see [MM00].

Given $\alpha,\beta\in{\mathcal{C}}^{s}_{0}(\dot{S})$ and a witness $W$, we define the distance between $\alpha$ and $\beta$ in $W$ by

Note that if $W=\dot{S}$, then $d_{\dot{S}}(\alpha,\beta)$ is simply the usual distance between $\alpha$ and $\beta$ in ${\mathcal{C}}(\dot{S})$. According to [MM00, Lemma 2.3], projections satisfy a $2$–Lipschitz projection bound.

We should mention that in [MM00] Masur and Minsky consider the map from ${\mathcal{C}}(\dot{S})$ and proves the second statement. Since ${\mathcal{C}}^{s}(\dot{S})$ is a subcomplex of ${\mathcal{C}}(\dot{S})$ for which every curve has non-empty projection, the first statement follows from the second.

We will also need the following important fact about projections from [MM00].

We assume (as we may) that $M\geq 8$, as this makes some of our estimates cleaner. In fact, there is a uniform $M$ that is independent of $Y$ in this theorem, given by Webb [Web15].

Consider the forgetful map

induced from the inclusion $\dot{S}\rightarrow S$. By definition of ${\mathcal{C}^{s}(\dot{S})}$, $\Pi$ is well defined since every curve in ${\mathcal{C}^{s}(\dot{S})}$ determines a curve in ${\mathcal{C}}(S)$. Each point $v\in{\mathcal{C}}(S)$ determines a weighted multi-curve: $v$ is contained in the interior of a unique simplex, which corresponds to a multi-curve on $S$, and the barycentric coordinates determine weights on the components of the multi-curve. According to [KLS09], the fiber of the map $\Pi$ is naturally identified with the Bass-Serre tree associated to the corresponding weighted multi-curve: $\Pi^{-1}(v)=T_{v}$.

An important tool in our analysis is the survival map

The construction of the analogous map when $S$ is closed is described in [LMS11]. Since there are no real subtleties that arise, we describe enough of the details of the construction for our purposes, and refer the reader to that paper for details. Before getting to the precise definition of $\Phi$, we note that for every $v\in{\mathcal{C}}(S)$, the restriction of $\Phi$ to $\mathbb{H}\cong\{v\}\times\mathbb{H}$ will be denoted $\Phi_{v}\colon\mathbb{H}\to{\mathcal{C}}^{s}(\dot{S})$, and this is simpler to describe: $\Phi_{v}$ is a $\pi_{1}S$–equivariantly factors as $\mathbb{H}\to T_{v}\to\Pi^{-1}(v)$, where the action of $\pi_{1}S$ on $\mathbb{H}$ comes from our reference hyperbolic structure on $S$, the associated covering map $p\colon\mathbb{H}\to S$, and choice of basepoint $\widetilde{z}\in p^{-1}(z)$.

To describe $\Phi$ in general, it is convenient to construct a more natural map from which $\Phi$ is defined as the descent to a quotient. Specifically, we will define a map

where $\operatorname{Diff}_{0}(S)$ is the component of the group of diffeomorphisms that of $S$ containing the identity (all diffeomorphisms of $S$ are assumed to extend to to diffeomorphisms of the closed surface obtained by filling in the punctures). To define $\widetilde{\Phi}$, first for each curve $\alpha\in{\mathcal{C}}_{0}(S)$, we let $\alpha$ denote the geodesic representative in our fixed hyperbolic metric on $S$, and choose once and for all $\epsilon(\alpha)>0$ so that for any two vertices $\alpha,\alpha^{\prime}$, $i(\alpha,\alpha^{\prime})$ is equal to the number of components of $N_{\epsilon(\alpha)}(\alpha)\cap N_{\epsilon(\alpha^{\prime})}(\alpha^{\prime})$. If $f(z)$ is disjoint from the interior of $N_{\epsilon(\alpha)}(\alpha)$, then $\widetilde{\Phi}(\alpha,f)=f^{-1}(\alpha)$, viewed as a curve on $\dot{S}$. If $f(z)$ is contained in the interior of $N_{\epsilon(\alpha)}(\alpha)$, then we let $\alpha_{\pm}$ denote the two boundary components of this neighborhood, and define $\widetilde{\Phi}(\alpha,f)$ to be a point of the edge between the curves $f^{-1}(\alpha_{-})$ and $f^{-1}(\alpha_{+})$ determined by the relative distance to $\alpha_{+}$ and $\alpha_{-}$. For a point $s\in{\mathcal{C}}(S)$ inside a simplex $\Delta$ of dimension greater than $0$, we use the neighborhoods as well as the barycentric coordinates of $s$ inside $\Delta$ to define $\widetilde{\Phi}(s,f)\in\Pi^{-1}(s)\subset\Pi^{-1}(\Delta)$; see [LMS11, Section 2.2] for details.

Next we note that the isotopy from the identity to $f$ lifts to an isotopy from the identity to a canonical lift $\widetilde{f}$ of $f$. The map $\Phi$ is then defined from our choice $\widetilde{z}\in p^{-1}(z)$ and the canonical lift by the equation

Alternatively, the we have the evaluation map ${\rm{ev}}\colon\operatorname{Diff}_{0}(S)\to S$, ${\rm{ev}}(f)=f(z)$, which lifts to a map $\widetilde{{\rm{ev}}}\colon\operatorname{Diff}_{0}(S)\to\mathbb{H}$ (given by $\widetilde{{\rm{ev}}}(f)=\widetilde{f}(\widetilde{z})$, where again $\widetilde{f}$ is the canonical lift), and then $\Phi$ is defined as the descent by ${\rm{id}}_{{\mathcal{C}}(S)}\times\widetilde{{\rm{ev}}}$:

Note that every $\widetilde{w}\in\mathbb{H}$ is $\widetilde{w}=\widetilde{f}(\widetilde{z})$ for some $f\in\operatorname{Diff}_{0}(S)$ (indeed, $\widetilde{{\rm{ev}}}$ defines a locally trivial fiber bundle). As is shown in [LMS11], $\Phi(\alpha,\widetilde{w})$ is well-defined independent of the choice of such a diffeomorphism $f$ with $\widetilde{f}(\widetilde{z})=\widetilde{w}$ since any two differ by an isotopy fixing $z$, and $\Phi$ is $\pi_{1}S$–equivariant (where the points $\widetilde{z}$ is used to identify the fundamental group with the group of covering transformations). It is straightforward to see that $\Phi(\alpha,\cdot)$ is constant on components of $\mathbb{H}{\smallsetminus}p^{-1}(N_{\epsilon(\alpha)}(\alpha))$: two points $\widetilde{w},\widetilde{w}^{\prime}$ in such a component are given by $\widetilde{w}=\widetilde{f}(\widetilde{z})$ and $\widetilde{w}^{\prime}=\widetilde{f}^{\prime}(\widetilde{z})$ where $f$ and $f^{\prime}$ are isotopic by an isotopy $f_{t}$, so that $f_{t}(z)$ remains outside $N_{\epsilon(\alpha)}(\alpha)$ for all $t$.

The following lemma relates $\Phi$ to the proper witnesses. Let $\mathcal{P}\subset\partial\mathbb{H}$ denote the set of parabolic fixed points. Assume that for each $x\in\mathcal{P}$, we choose a horoball $H_{x}$ invariant by the parabolic subgroup ${\rm{Stab}}_{\pi_{1}S}(x)$, the stabilizer of $x$ in $\pi_{1}S$. We further assume, as we may, that (1) the union of the horoballs is $\pi_{1}S$–invariant, (2) the horoballs are pairwise disjoint (so all projected to pairwise disjoint cusp neighborhoods of the punctures), and (3) the horoballs all project disjoint from $N_{\epsilon(\alpha)}(\alpha)$ for all curves $\alpha$. Recall that any proper witness is determined by its boundary.

From the lemma (and as illustrated in the proof) $\Phi|_{C(S)\times H_{x}}$ defines an isomorphism ${\mathcal{C}}(S)\to{\mathcal{C}}(\mathcal{W}(x))$ inverting the isomorphism $\Pi|_{{\mathcal{C}}(\mathcal{W}(x))}\colon{\mathcal{C}}(\mathcal{W}(x))\to{\mathcal{C}}(S)$.

We refer the reader to [Thu88], [CEG06], [FLP12], and [CB88] for details about the topics discussed here. By a lamination on a surface $Y$ we mean a compact subset of the interior of $Y$ foliated by complete geodesics with respect to some complete, hyperbolic metric of finite area, with (possibly empty) geodesic boundary; the geodesics in the foliation are uniquely determined by the lamination and are called the leaves. For example, any simple closed geodesic $\alpha$ is a lamination with exactly one leaf. For a fixed complete, finite area, hyperbolic metric on $Y$, all geodesic laminations are all contained in a compact subset of the interior of $Y$. For any two complete, hyperbolic metrics of finite area, laminations that are geodesic with respect to the first are isotopic to laminations that are geodesic with respect to the second. In fact, we can remove any geodesic boundary components, and replace the resulting ends with cusps, and this remains true. We therefore sometimes view laminations as well-defined up to isotopy, unless a hyperbolic metric is specified in which case we assume they are geodesic.

A complementary region of a lamination $\mathcal{L}\subset Y$ is the image in $Y$ of the closure of a component of the preimage in the universal covering; intuitively, it is the union of a complementary component together with the ``leaves bounding this component''. We view the complementary regions as immersed subsurfaces with (not necessarily compact) boundary consisting of arcs and circles (for a generic lamination, the immersion is injective, though in general it is only injective on the interior of the subsurface). We will also refer to the closure of a complementary component in the universal cover of $Y$ as a complementary region (of the preimage of a lamination).

We write $\mathcal{G}\mathcal{L}(Y)$ for the set of laminations on the surface $Y$, dropping the reference to $Y$ when it is clear from the context. The set of essential simple closed curves, up to isotopy (i.e. the vertex set of ${\mathcal{C}}(Y)$) is thus naturally a subset of $\mathcal{G}\mathcal{L}(Y)$. A lamination is minimal if every leaf is dense in it, and it is filling if its complementary regions are ideal polygons, or one-holed ideal polygons where the hole is either a boundary component or cusp of $Y$. A sublamination of a lamination is a subset which is also a lamination. Every lamination decomposes as a finite disjoint union of simple closed curves, minimal sublaminations without closed leaves (called the minimal components), and biinfinite isolated leaves (leaves with a neighborhood disjoint from the rest of the lamination).

There are several topologies on $\mathcal{G}\mathcal{L}$ that will be important for us (in what follows, and whenever discussing convergence in the topologies, we view laminations as geodesic laminations with respect to a fixed complete hyperbolic metric of finite area; the resulting topology and convergence is independent of the choice of metric). The first is a metric topology called the Hausdorff topology (also known as the Chabauty topology), induced by the Hausdorff metric on the set of all compact subsets of a compact space (in our case, the compact subset of the surface that contains all geodesic laminations) defined by

If a sequence of $\{\mathcal{L}_{i}\}$ converges to $\mathcal{L}$ in this topology, we write $\mathcal{L}_{i}\xrightarrow{\text{H}}\mathcal{L}$. The following provides a useful characterization of convergence in this topology; see [CEG06].

This lemma holds not just for Hausdorff convergence of laminations, but for any sequence of compact subsets of a compact metric space with respect to the Hausdorff metric.

The set $\mathcal{G}\mathcal{L}$ can also be equipped with a weaker topology called the coarse Hausdorff topology, [Ham06], introduced by Thurston in [Thu78] where it was called the geometric topology (see also [CEG06] where it was Thurston topology). If a sequence $\{\mathcal{L}_{i}\}$ converges to $\mathcal{L}$ in the coarse Hausdorff topology, then we write $\mathcal{L}_{i}\xrightarrow{\text{CH}}\mathcal{L}$. The following describes convergence in this topology; see [CEG06].

The next corollary gives a useful way of understanding coarse Hausdorff convergence.

Since any lamination has only finitely many sublaminations, from the corollary we see that while limits are not necessarily unique in the coarse Hausdorff topology, a sequence can have only finitely many limits. We let $\mathcal{EL}=\mathcal{EL}(Y)$ denote the space of ending laminations on $Y$, which are minimal, filling laminations, equipped with the coarse Hausdorff topology. As suggested by the name, these are precisely the laminations that occur as the ending laminations of a type preserving, proper, isometric action on hyperbolic $3$–space without accidental parabolics as discussed in the introduction.

A measured lamination is a lamination $\mathcal{L}$ together with an invariant transverse measure $\mu$; that is, an assignment of a measure on all arcs transverse to the lamination, satisfying natural subdivision properties which is invariant under isotopy of arcs preserving transversality with the lamination. The support of a measured lamination $(\mathcal{L},\mu)$ is the sublamination $|\mu|\subseteq\mathcal{L}$ with the property that a transverse arc has positive measure if and only if the intersection with $|\mu|$ is nonempty, and is a union of minimal components and simple closed geodesics. We often assume that $(\mathcal{L},\mu)$ has full support, meaning $\mathcal{L}=|\mu|$. In this case, we sometimes write $\mu$ instead of $(\mathcal{L},\mu)$.

The space $\mathcal{ML}=\mathcal{ML}(Y)$ of measured laminations on $Y$ is the set of all measured laminations of full support equipped with the weak* topology on measures on an appropriate family of arcs transverse to all laminations. Given an arbitrary measured lamination, $(\mathcal{L},\mu)$, we have $(|\mu|,\mu)$ is an element of $\mathcal{ML}$, and so every measured lamination determines a unique point of $\mathcal{ML}$. We let $\mathcal{FL}\subset\mathcal{ML}$ denote the subspace of measured laminations whose support is an ending lamination (i.e. it is in $\mathcal{EL}$). We write $\mathcal{PML}$ and $\mathcal{PFL}$ for the respective projectivizations of $\mathcal{ML}$ and $\mathcal{FL}$, obtained by taking the quotient by scaling measures, with the quotient topologies. The following will be useful in the sequel; see [Thu78, Chapter 8.10].

For the surface $\dot{S}$, we consider the subspace

which is the union of ending laminations of all witnesses of $\dot{S}$. Similarly, we will write $\mathcal{FL}^{s}(\dot{S})\subset\mathcal{ML}(\dot{S})$ for those measured laminations supported on laminations in $\mathcal{E}\mathcal{L}^{s}(\dot{S})$, and $\mathcal{PFL}^{s}(\dot{S})\subset\mathcal{PML}(\dot{S})$ for its projectivization.

A $\delta$–hyperbolic space $\mathcal{X}$ can be equipped with a boundary at infinity, $\partial\mathcal{X}$ as follows. Given $x,y\in\mathcal{X}$ and a basepoint $o\in\mathcal{X}$, the Gromov product of $x$ and $y$ based at $o$ is given by

Up to a bounded error (depending only on $\delta$), $\langle x,y\rangle_{o}$ is the distance from $o$ to a geodesic connecting $x$ and $y$. The quantity $\langle x,y\rangle_{o}$ is estimated by the distance from the basepoint $o$ to a quasi-geodesic between $x$ and $y$. There is an additive and multiplicative error in the estimate that depends only on the hyperbolicity constant and the quasi-geodesic constants. Using and slim triangles, we also note that for all $x,y,z\in\mathcal{X}$,

where the constants in the coarse lower bound depend only on the hyperbolicity constant.

A sequence $\{x_{n}\}\subset\mathcal{X}$ is said to converge to infinity if $\displaystyle{\lim_{m,n\rightarrow\infty}\langle x_{m},x_{n}\rangle_{o}=\infty}$. Two sequences $\{x_{n}\}$ and $\{y_{n}\}$ are equivalent if $\displaystyle{\lim_{m,n\rightarrow\infty}\langle y_{m},x_{n}\rangle=\infty}$. The points in $\partial\mathcal{X}$ are equivalence classes of sequences converging to infinity, and if $\{x_{k}\}\in x\in\partial\mathcal{X}$, then we say $\{x_{k}\}$ converges to $x$ and write $x_{k}\to x$ in $\overline{\mathcal{X}}=\mathcal{X}\cup\partial\mathcal{X}$. The topology on the boundary is such that a sequence $\{x^{n}\}_{n}\subset\partial X$ converges to a point $x\in\partial X$ if there exist sequences $\{x_{k}^{n}\}_{k}$ representing $x^{n}$ for all $n$, and $\{x_{m}\}_{m}$ representing $x$ so that

For details see, e.g. [BH99, KB02].