2. Relative hyperbolicity à la Groves and Manning

We denote by $\Lambda \times \{k\}$ the subgraph of $\mathcal{H}(\Lambda)$ spanned by the vertex set $\Lambda^{(0)} \times \{k\}$ .

The following definition is due to Groves and Manning, who prove that it is equivalent to strong relative hyperbolicity [ Reference FarbFar98 , Reference BowditchBow12 ]. We refer the reader to [ Reference Groves and ManningGM08 , theorem 3·25] for a proof and more details. A detailed study and equivalences of various notions of relative hyperbolicity was done by Hruska in [ Reference HruskaHru10 ].

3. Horoballs and convexity of parabolics

It is well known that given a relatively hyperbolic group, its parabolic subgroups are quasiconvex [ Reference Druţu and SapirDS05 , Lemma 4·15]. The goal of this section is to prove Theorem 3·5, which says that a relatively hyperbolic graph can be modified so that its parabolic subgraphs are convex subgraphs. We make use of several previously known results.

Lemma 3·3 is essentially a re-statement of Lemma 3·10 of [ Reference Groves and ManningGM08 ] in the context of restricted horoballs, and our proof below, given for the sake of completeness, is almost identical to theirs.

Let us first make the convention that a vertical segment of a path $\gamma$ is a subpath whose image is the union of vertical edges in a horoball. Similarly a horizontal segment is a subpath whose image is disjoint from the set of vertical edges.

Proof. We start the proof with a basic observation. Let $1 \leq m < n$ and let (x, m) and (y, m) be two points in $\Lambda\times \{m\}$ . If (x, m) and (y, m) are at ( $\Lambda\times \{m\}$ )-distance D, note that the ( $\Lambda\times \{m+1\}$ )-distance between $(x,m+1)$ and $(y,m+1)$ is $\bigl\lceil{D}/{2}\bigr\rceil$ (see Figure 1). Similarly, the ( $\Lambda\times \{m+k\}$ )-distance between $(x,m+k)$ and $(y,m+k)$ is $\bigl\lceil{D}/{2^k}\bigr\rceil$ . This observation implies the following:

1. (1) Assume that a geodesic path contains a horizontal segment in $\Lambda\times \{m\}$ of length more than one. Assume that this horizontal segment is not contained in a strictly larger horizontal segment of the geodesic. Then the vertical segment immediately preceding the horizontal segment is an ascending segment, in the sense that it is a vertical segment from some $\Lambda\times \{m-k\}$ to $\Lambda\times \{m\}$ . Similarly, the immediate successor of the horizontal segment is a descending segment. See Figure 2 for an illustration.

2. (2) Any geodesic path with a descending segment at (x, m) cannot ascend back to $\Lambda\times \{m\}$ in the future (see Figure 3). In other words, no ascending segment follows a descending segment.

3. (3) Any geodesic path contains at most two maximal descending (respectively ascending) segments. See Figure 4.

Fig. 1.

The ( $\Lambda\times \{m\}$ )-distance between (x, m) and (y, m) is 8 while the ( $\Lambda\times \{m+1\}$ )-distance between $(x,m+1)$ and $(y,m+1)$ is 4.

Fig. 2.

The thickened path between (x, m) and $(y,m+1)$ on the left is longer than the thickened path on the right.

Fig. 3.

The thickened path between (x, m) and (y, m) on the bottom panel is shorter than the one on the top panel.

Fig. 4.

The thickened path between (x, m) and $(y,m-3)$ on the left is longer than the one on the right.

Let $\gamma$ be a geodesic between the points $v_1$ and $v_2$ in the statement. By the above observations, if $\gamma$ contains a horizontal segment of length at least two at some $\Lambda\times \{m\}$ , then $\text{im}(\gamma)$ is disjoint from $\Lambda\times \{m^{\prime}\}$ for all $m < m^{\prime} \leq n$ . Thus, any horizontal segment in $\gamma$ is either of length one, or is contained in the maximum level $\Lambda\times \{\max\}$ that intersects $\text{im}(\gamma)$ nontrivially.

In fact, it can be verified that apart from the horizontal segment at $\Lambda\times \{\max\}$ , the image of $\gamma$ can have at most one more horizontal edge.

Another consequence of the above observations is that if $\gamma$ contains a horizontal segment of length at least 6, then this segment has to be contained in $\Lambda\times \{n\}$ , see Figure 5.

Fig. 5.

If $m <n$ , then the thickened horizontal path between (x, m) and (y, m) in the top panel is longer than the red path in the bottom panel.

Assume that the horizontal edge not at $\Lambda \times \{\max\}$ is an edge between (x, m) and (y, m) and is followed by an ascending segment from (y, m) to $(y,\max\!) \in \Lambda\times \{\max\}$ . Let $\gamma^{\prime}$ be the geodesic obtained from $\gamma$ by replacing the above by a vertical segment from (x, m) to $(x,\max\!)$ followed by a horizontal edge to $(y,\max\!)$ . If $\max < n$ and the only horizontal segment of $\gamma^{\prime}$ contains 4 or 5 edges, then let $\beta$ be the geodesic obtained by replacing this horizontal segment by an ascending edge, a horizontal segment in $\Lambda\times \{\max+1\}$ and a descending edge back to $\Lambda\times \{\max\}$ , similar to the procedure in Figure 5. We leave it as an exercise to verify that $\beta$ is as required.

Before stating the main result of this section, we recall a convexity result from [ Reference Groves and ManningGM08 ] which will be used in the proof.

Proof. Let $\delta$ be the hyperbolicity constant of $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ . Let $r = \lceil 8\delta+2 \rceil$ and $n \ge 2r + M(\delta, 2\delta, K)$ , where K is the constant from Lemma 3·1 and M is the constant from Theorem 3·2 Fix $\alpha_0 \in \mathscr{A}$ and points $x,y \in \Lambda_{\alpha_0}\times\{n\}$ . Let $\gamma \,:\, P \to \mathcal{H}_n \bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ be a geodesic (in $\mathcal{H}_n \bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ ) between x and y. Since each n-restricted horoball in $\mathcal{H}_n \bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is a full subgraph, every subpath of $\gamma$ whose image lies in an n-restricted horoball is a geodesic in that horoball. We will therefore assume that each such geodesic subpath of $\gamma$ is of the form given by Lemma 3·3.

Denote by $U \subset \mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ the set $\bigcup_{\alpha \in \mathscr{A}} N_r \bigl(\Lambda_{\alpha}\times \{n\}\bigr)$ . The path $\gamma$ is a concatenation $\gamma_1 \cdot \beta_1\cdot \gamma_2\cdots \gamma_k$ , where each $\gamma_i$ is a path with image in U and each $\beta_i$ is such that its image is disjoint from U, except at the endpoints. See Figure 6 for an illustration.

Fig. 6.

The path $\gamma$ is a concatenation of the paths $\gamma_i$ (geodesics that lie in the r-neighbourhoods of the parabolics) and $\beta_j$ (geodesics between the $\gamma_i$ ).

Note that by Lemma 3·3, each $\beta_i$ is a path which satisfies the following:

1. (i) $\text{im}(\beta_i)$ is not contained in any single n-restricted horoball and thus has length at least $2(n-r) > 2r$ , and

2. (ii) for any $\alpha \in \mathscr{A}$ , $\text{im}(\beta_i) \cap \mathcal{H}_n\bigl(\Lambda_{\alpha}\bigr)$ is a union of components, where each component is either a vertical segment between $\Lambda_{\alpha}\times \{0\}$ and $\Lambda_{\alpha}\times \{n-r\}$ (e.g., $\text{im}(\beta_2) \cap \mathcal{H}_n\bigl(\Lambda_{\alpha_2}\bigr)$ in Figure 6), or the image of a geodesic between points of $\Lambda_{\alpha}\times \{0\}$ (e.g., $\text{im}(\beta_1) \cap \mathcal{H}_n\bigl(\Lambda_{\alpha_1}\bigr)$ in Figure 6). In the latter case, we note that this component is disjoint from the image of any $\gamma_j$ .

Let $\iota \,:\, \mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr) \hookrightarrow \mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ denote the inclusion map. For each i, let $\gamma^{\prime}_i$ be a geodesic path in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ between the endpoints of $\iota \circ \gamma_i$ . Let $\gamma^{\prime} \,:\, P^{\prime} \to \mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ be the path obtained from $\iota \circ \gamma$ by replacing each $\iota \circ \gamma_i$ by $\gamma^{\prime}_i$ . We will denote $\iota \circ \beta_i$ by $\beta^{\prime}_i$ . Thus $\gamma^{\prime} = \gamma^{\prime}_1 \cdot \beta^{\prime}_1 \cdot \gamma^{\prime}_2 \cdots \gamma^{\prime}_k$ .

Claim. The path $\gamma^{\prime}$ is an r-local geodesic in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ .

Proof. Each $\gamma^{\prime}_i$ is a geodesic, and therefore a local geodesic. Each $\beta^{\prime}_i$ is an r-local geodesic since the r-ball around any point in $\text{im}(\beta_i)$ is contained in $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ .

As observed above, the image of every subpath of $\beta^{\prime}_i$ that lies in a horoball is either a vertical segment or it does not meet any $\gamma^{\prime}_j$ . This implies that any subpath of $\beta^{\prime}_{i-1}\cdot\gamma^{\prime}_i\cdot\beta^{\prime}_i$ whose image lies in $\mathcal{H}\bigl(\Lambda_{\alpha}\bigr) \setminus \mathcal{H}_r\bigl(\Lambda_{\alpha}\bigr)$ is a geodesic in $\mathcal{H}\bigl(\Lambda_{\alpha}\bigr) \setminus \mathcal{H}_r\bigl(\Lambda_{\alpha}\bigr)$ . Since $\mathcal{H}\bigl(\Lambda_{\alpha}\bigr) \setminus \mathcal{H}_r\bigl(\Lambda_{\alpha}\bigr)$ is convex (by Lemma 3·4), each such subpath is in fact a geodesic in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ , and therefore an r-local geodesic. This proves the claim.

Thus by Lemma 3·1, $\gamma^{\prime}$ is a $(2\delta,K)$ -quasi-geodesic and by Theorem 3·2, it lies in an $M = M(\delta, 2\delta, K)$ neighbourhood of any geodesic in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ between x and y. Since $x,y \in \Lambda_{\alpha_0}\times \{n\}$ with $n - 1 > \delta$ , we have that any geodesic between them in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ lies in $\mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr)\setminus \mathcal{H}_{n-1}\bigl(\Lambda_{\alpha_0}\bigr)$ (again, by Lemma 3·4). This implies that $\gamma^{\prime}$ lies in $N_M(\mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr) \setminus \mathcal{H}_{n-1}\bigl(\Lambda_{\alpha_0}\bigr)) \subset \mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr) \setminus \mathcal{H}_{2r}\bigl(\Lambda_{\alpha_0}\bigr)$ .

We are thus forced to conclude that $\gamma^{\prime} = \gamma^{\prime}_1$ (and therefore $\gamma = \gamma_1$ ). Indeed, if not, then $\beta_1$ is a geodesic in $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ with endpoints on $\Lambda_{\alpha_0}\times\{n-r\}$ and such that $\text{im}(\beta_1) \subset \mathcal{H}_n\bigl(\Lambda_{\alpha_0}\bigr)$ . But as observed above, $\text{im}(\beta_1)$ is not contained in any single n-restricted horoball, which is a contradiction.

Using Lemma 3·3 once again, we conclude that $\gamma \subset \Lambda_{\alpha_0}\times \{n\}$ .

Proof. Let $(x,0), (y,0) \in \Lambda_{\alpha}\times \{0\}$ be vertices at the bottom level of the combinatorial horoball based on $\Lambda_{\alpha}$ in $\Gamma$ and let $(x,n), (y,n) \in \Lambda_{\alpha}\times \{n\}$ be the corresponding vertices at the nth level. We have

\[ \Bigl| d_{\mathcal{H}_n}\bigl((x,0),(y,0)\bigr) - d_{\mathcal{H}_n}\bigl((x,n),(y,n)\bigr) \Bigr| \le 2n \]

by the triangle inequality. It follows that the map $(\Lambda_{\alpha}^{(0)} \times {0}, d_{\mathcal{H}_n}) \to (\Lambda_{\alpha}^{(0)} \times \{n\}, d_{\mathcal{H}_n})$ given by $(x,0) \mapsto (x,n)$ is a rough isometry and $\Lambda_{\alpha}^{(0)} \times \{n\}$ is a convex subgraph of $\mathcal{H}_n$ .

We now recall and prove Theorem 1·3.

Proof. Let S be a finite generating set of G containing each of the $S_i$ . Let $\Gamma$ be the Cayley graph of G with respect to S. Then the Cayley graphs $\Gamma^{\prime}_i = \text{Cayley}(H_i,S_i)$ are subgraphs of $\Gamma$ and G is hyperbolic relative to the family $(g\Gamma^{\prime}_i)_{g,i}$ of G-translates of these subgraphs. By Theorem 3·5, there is an n for which the parabolics of $\mathcal{H}_n\bigl(\Gamma, (g\Gamma^{\prime}_i)_{g,i}\bigr)$ are convex. For each i, let $\Gamma_i$ be the parabolic in the restricted horoball with base $\Gamma^{\prime}_i$ . Then $\mathcal{H}_n$ and the $\Gamma_i$ satisfy all the required conditions.

4. A Cayley graph with strongly shortcut parabolics

Let G be a finitely generated group that is strongly shortcut relative to strongly shortcut subgroups $(H_i)_i$ . In this section we will show that there exists a generating set S for G such that the $H_i$ are strongly shortcut metric subspaces of the Cayley graph $\text{Cayley}(G,S)$ . In order to do this, we will first need to define what it means for a metric space to be strongly shortcut. The following definition appears in earlier work of the first named author under the name nonapproximability of n-gons [ Reference HodaHod20 , definition 3·2].

Our aim in this section is to prove the following.

In order to prove Theorem 4·3 we will rely on Theorem 3·5 and the following refined version of the Milnor–Švarc Lemma. This version of the Milnor–Švarc Lemma gives us arbitrary control on the multiplicative constant of the quasi-isometry, up to scaling the metric on the Cayley graph. This arbitrary control on the multiplicative constant of the quasi-isometry comes at the cost of having to choose larger and larger finite generating sets and accepting larger and larger additive quasi-isometry constants.

Theorem 4·4 (Fine Milnor–Švarc Lemma [ Reference HodaHod20 , theorem H]). Let (X,d) be a geodesic metric space. Let G be a group acting metrically properly and coboundedly on X by isometries. Fix $x_0 \in X$ . For $t > 0$ let $S_t$ be the finite set defined by

\[ S_t = \bigl\{ g \in G \,:\, d(x_0,gx_0) \le t \bigr\} \]

and consider the word metric $d_{S_t}$ defined by $S_t$ . (For those t where $S_t$ does not generate G, we allow $d_{S_t}$ to take the value $\infty$ ). Let $K_t$ be the infimum of all $K > 1$ for which

\begin{align*} (G,td_{S_t}) & \longrightarrow X \\ g & \longmapsto g \cdot x_0 \end{align*}

is a $(K, C_K)$ -quasi-isometry for some $C_K \ge 0 $ . Then $K_t \to 1$ as $t \to \infty$ .

Lemma 4·5. Let G be a finitely generated group that is hyperbolic relative to finitely generated subgroups $(H_i)_i$ . For each i, let $S_i$ be a finite generating set for $H_i$ . Then, for any $L > 1$ , there is a $t > 0$ and a finite generating set S for G such that each inclusion

\[ (H_i, d_{S_i}) \hookrightarrow (G, td_S) \]

is a quasi-isometric embedding with multiplicative constant L, where $d_S$ and the $d_{S_i}$ are the word metrics.

Proof. Let $S^{\prime} \supseteq \bigcup_i S_i$ be a finite generating set for G. Let $\Gamma^{\prime} = \text{Cayley}(G,S^{\prime})$ and let $\Lambda_{g,i} = g\text{Cayley}(H_i,S_i)$ . By Theorem 3·5, for some n, the top level subgraphs $\Lambda_{g,i}\times \{n\}$ of the restricted horoballs of $\mathcal{H}_n = \mathcal{H}_n\bigl(\Gamma^{\prime},\{\Lambda_{g,i}\}_{g,i}\bigr)$ are convex. Moreover, by Remark 2·9, the group G acts properly and cocompactly on $\mathcal{H}_n$ .

By Corollary 3·6 and Remark 2·4, there is a rough isometry $(\Lambda_{e,i}^{(0)}\times \{0\}, d_{\mathcal{H}_n}) \to \bigl(H_i, {1}/{2^n}d_{S_i}\bigr)$ . By Theorem 4·4, there is a generating set S for G and a scaling factor $t^{\prime} > 0$ such that the inclusion $(G, t^{\prime} d_S) \hookrightarrow \mathcal{H}_n$ is a quasi-isometry with multiplicative constant L, where $d_S$ is the word metric coming from S. But the image of $H_i$ under this inclusion is $\Lambda_{e,i}^{(0)}\times \{0\}$ and so the composition of the restriction $(H_i, t^{\prime} d_S) \hookrightarrow (\Lambda_{e,i}^{(0)}\times \{0\}, d_{\mathcal{H}_n})$ and the rough isometry $(\Lambda_{e,i}^{(0)}\times \{0\}, d_{\mathcal{H}_n}) \to \bigl(H_i, {1}/{2^n}d_{S_i}\bigr)$ gives us a quasi-isometry $(H_i, t^{\prime} d_S) \to \bigl(H_i, {1}/{2^n}d_{S_i}\bigr)$ with multiplicative constant L. Scaling the domain and the codomain by $2^n$ , taking the quasi-inverse and composing it with the isometric embedding $(H_i, 2^n t^{\prime} d_S) \hookrightarrow (G, 2^n t^{\prime} d_S)$ we obtain a quasi-isometry $(H_i, d_{S_i}) \hookrightarrow (G, 2^n t^{\prime} d_S)$ with multiplicative factor L.

Finally, we will need the next two theorems about strongly shortcut spaces and groups.

Proof of Theorem 4·3. Let G be a finitely generated group that is hyperbolic relative to strongly shortcut groups $(H_i)_i$ . By Theorem 4·7, we can choose finite generating sets $S_i$ of $H_i$ so that the Cayley graphs $\text{Cayley}(H_i,S_i)$ are strongly shortcut. Then, by Theorem 4·6, for each i, there exists an $L_i > 1$ such that any metric space that, up to scaling, is quasi-isometric to $(H_i, d_{S_i})$ with multiplicative constant $L_i$ is also strongly shortcut. By Lemma 4·5, there is a finite generating set S of G and a $t > 0$ such that, for each i, if $d_S$ is the word metric coming from S then $(H_i, t d_S)$ is quasi-isometric to $(H_i, d_{S_i})$ with multiplicative constant $L = \min_i L_i$ . Thus each $(H_i, d_S)$ is strongly shortcut.

5. Asymptotic cones and the proof of the main result

In this section we will recall the definition of asymptotic cones of metric spaces. Then we will state the theorem of Osin and Sapir on tree-gradedness of asymptotic cones of relatively hyperbolic groups and a theorem of the second named author giving an asymptotic cone characterization of the strong shortcut property. We will use these theorems and the results of the previous sections to prove Theorem 1·2.

For an exposition of asymptotic cones, see Drutu and Kapovich [ Reference Druţu and KapovichDK18 ].

The existence of non-principal ultrafilters is a consequence of Zorn’s Lemma (see [ Reference Druţu and KapovichDK18 , lemma 10·18] for instance).

Let (X, d) be a metric space and let $\omega$ be a non-principal ultrafilter over $\mathbb{N}$ . Let $(r_n)_n$ be a sequence of real numbers such that $\lim_{\omega} r_n = \infty$ . Fix a sequence of basepoints $(p_n)_n \in X^{\mathbb{N}}$ .

Let $d_{\infty} \,:\, X^{\mathbb{N}} \times X^{\mathbb{N}} \to [0,\infty]$ be defined as $d_{\infty} ((x_n)_n, (y_n)_n) \,:\!=\, \lim_{\omega} (d(x_n,y_n)/r_n)$ . Let $X_B^{\mathbb{N}}((r_n)_n,(p_n)_n) \,:\!=\, \{(x_n)_n \in X^{\mathbb{N}} | d_{\infty}((x_n)_n,(p_n)_n) < \infty\}$ .

Theorem 5·8 (Osin and Sapir [ Reference Druţu and SapirDS05 , theorem A·1]). Let G be a finitely generated group and let $d_S$ be a word metric coming from a finite generating set S of G. If G is hyperbolic relative to a family of subgroups $(H_i)_i$ then every asymptotic cone $\mathcal{A} = \text{Cone}_{\omega}\bigl((G,d_S),(r_n)\bigr)$ of $(G, d_S)$ is tree graded with respect to the $\omega$ -limits

\[ \lim_{\omega} (g_n H_i)_n = \bigl\{ [x_n]_n \in \mathcal{A} \,:\, x_n \in g_n H_i \bigr\} \]

of the $(g_n H_i)_n$ with $[g_n]_n \in \mathcal{A}$ .

Remark 5·9. The $\lim_{\omega} (g_n H_i)_n$ are isometric to asymptotic cones of the $(H_i, d_S)$ . Indeed, the asymptotic cone $\mathcal{A}$ is a group with multiplication given by

\[ [x_n]_n \cdot [y_n]_n = [x_ny_n]_n \]

and $d_{\infty}$ is a left-invariant metric with respect to this group structure. Thus $\lim_{\omega} (g_n H_i)_n$ is isometric to

\[ [g_n^{-1}]_n\lim_{\omega} (g_n H_i)_n = \lim_{\omega} (H_i)_n \]

which is $\text{Cone}_{\omega}\bigl((H_i,d_S),(r_n)\bigr)$ .

A Riemannian circle C is $S^1$ equipped with a geodesic metric of some length $|C|$ . In other words C is the quotient of $\mathbb{R}$ by the action of $|C|\mathbb{Z}$ .

We are now ready to prove our main result, which we first recall:

Proof. Let G be a finitely generated group that is hyperbolic relative to strongly shortcut groups $(H_i)_i$ . By Theorem 4·3, there is a finite generating set S of G such that $(H_i, d_S)$ is strongly shortcut for each i, where $d_S$ is the word metric coming from S. We will show that the Cayley graph $\text{Cayley}(G,S)$ is strongly shortcut. By Theorem 4·2 and Theorem 5·10 it will suffice to prove that no asymptotic cone $\mathcal{A}$ of $\text{Cayley}(G,S)$ contains a Riemannian circle of unit length.

By Theorem 5·8, any embedded copy of C in $\mathcal{A}$ is contained in some $\lim_{\omega} (g_n H_i)_n$ with $[g_n]_n \in \mathcal{A}$ . Thus it suffices to show that $\lim_{\omega} (g_n H_i)_n$ does not contain an isometric copy of the Riemannian circle of unit length. But by Remark 5·9, the $\omega$ -limit $\lim_{\omega} (g_n H_i)_n$ is isometric to an asymptotic cone $\mathcal{A}^{\prime}$ of $(H_i, d_{S^{\prime}})$ , which is strongly shortcut. Hence $\mathcal{A}^{\prime}$ cannot contain an isometric copy of the Riemannian circle of unit length, by Theorem 5·10.