1. Introduction
Strongly shortcut graphs and groups were introduced by the first named author [ Reference HodaHod18 ] who later generalised the strong shortcut property to rough geodesic metric spaces [ Reference HodaHod20 ]. The strong shortcut property is a very general form of nonpositive curvature condition satisfied by many spaces of interest in geometric group theory, metric graph theory and geometric topology. These include Gromovhyperbolic spaces [ Reference HodaHod18 ], asymptotically $\text{CAT}(0)$ spaces [ Reference HodaHod20 ], hierarchically hyperbolic spaces, coarse Helly metric spaces of uniformly bounded geometry [ Reference Haettel, Hoda and PetytHHP20 ], 1skeletons of finite dimensional $\text{CAT}(0)$ cube complexes (i.e. median graphs), 1skeletons of quadric complexes (i.e. hereditary modular graphs), 1skeletons of systolic complexes (i.e. bridged graphs), standard Cayley graphs of Coxeter groups [ Reference HodaHod18 ] and all of the Thurston geometries except Sol [ Reference Hoda and PrzytyckiHP , Reference KarKar11 ]. Despite this surprisingly unifying nature, there are nonetheless important consequences for groups that act metrically properly and coboundedly on strongly shortcut geodesic metric spaces: finite presentability, polynomial isoperimetric function and thus decidable word problem [ Reference HodaHod18 , Reference HodaHod20 ].
The strong shortcut property is essentially about limitations on the scale and precision at which subspaces can approximate circles. Specifically:
Definition 1·1 (Strongly shortcut). A graph $\Gamma$ is strongly shortcut if, for some $K > 1$ there is a bound on the lengths of the Kbilipschitz combinatorial cycles in $\Gamma$ . A group G is strongly shortcut if G acts properly and cocompactly on a strongly shortcut graph.
This turns out to be equivalent to the existence of a metrically proper and cobounded Gaction on a strongly shortcut geodesic metric space, which we define in Section 4. Thus the following classes of groups are all strongly shortcut: hyperbolic groups [ Reference GromovGro87 ], asymptotically $\text{CAT}(0)$ groups [ Reference KarKar11 ] (e.g. $\text{CAT}(0)$ groups [ Reference Bridson and HaefligerBH99 ]), hierarchically hyperbolic groups [ Reference Behrstock, Hagen and SistoBHS17 , Reference Behrstock, Hagen and SistoBHS19 ] (e.g. mapping class groups of surfaces [ Reference Masur and MinskyMM99 , Reference Masur and MinskyMM00 ]), coarse Helly groups [ Reference Chalopin, Chepoi, Genevois, Hirai and OsajdaCCG+20 ] (e.g. Artin groups of FCtype, weak Garside groups [ Reference Huang and OsajdaHO19 ]), the discrete Heisenberg group [ Reference Hoda and PrzytyckiHP ], systolic groups (e.g. finitely presented C(6) small cancellation groups [ Reference WiseWis03 ]) and quadric groups (e.g. C(4)T(4) small cancellation groups) [ Reference HodaHod17 ].
Our main result is the following.
Theorem 1·2. Let G be a finitely generated group that is hyperbolic relative to strongly shortcut groups. Then G is strongly shortcut.
Theorem 1·2 allows us to obtain examples of strongly shortcut groups that are not known to be strongly shortcut by any other means. For example, let G be the free product of two copies of the discrete Heisenberg group and let $\langle t \rangle$ be a maximal cyclic subgroup generated by a loxodromic element t of the Bass–Serre tree of G. Then the amalgamated free product $G \ast_{\langle t \rangle} G$ is hyperbolic relative to discrete Heisenberg subgroups by Dahmani [ Reference DahmaniDah03 ] and thus is strongly shortcut by Theorem 1·2 and [ Reference Hoda and PrzytyckiHP ].
Our approach to proving Theorem 1·2 is to use properties of asymptotic cones of strongly shortcut groups and relatively hyperbolic groups. A result of the first named author characterises strongly shortcut groups as those whose asymptotic cones have no isometrically embedded circles ([ Reference HodaHod20 , theorem 3·7]), while a result of Osin and Sapir [ Reference Druţu and SapirDS05 , theorem A·1] guarantees that asymptotic cones of relatively hyperbolic groups are treegraded. Thus, any isometrically embedded circle in an asymptotic cone of a relatively hyperbolic group has to be contained in a piece, which is impossible if the peripherals are strongly shortcut.
In the course of the proof of Theorem 1·2 we restrict the combinatorial horoball construction of Groves and Manning [ Reference Groves and ManningGM08 ] to a sufficiently large finite number of levels, thus obtaining the following result which may be of independent interest.
Theorem 1·3. 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 there is a connected, free cocompact Ggraph $\Gamma$ with subgraphs $(\Gamma_i)_i$ such that, for each i:

(1) $\Gamma_i$ is a Rips graph of $\text{Cayley}(H_i,S_i)$ ;

(2) $H_i$ stabilises $\Gamma_i$ ;

(3) the $H_i$ action on $\Gamma_i$ is free and cocompact; and

(4) $\Gamma_i$ is convex in $\Gamma$ .
We use Theorem 1·3 to prove Theorem 4·3, which says that G has a Cayley graph in which the $H_i$ are strongly shortcut subspaces.
Structure of the paper. In Section 2, we recall the Groves and Manning combinatorial horoball construction and their characterisation of relative hyperbolicity. Section 3 is devoted to the proof of Theorem 1·3. In Section 4, we show that a relatively hyperbolic group with strongly shortcut parabolics admits a Cayley graph in which the parabolics are strongly shortcut subspaces. Finally, we recall the notion of asymptotic cones and prove the main result Theorem 1·2 in Section 5.
2. Relative hyperbolicity à la Groves and Manning
Definition 2·1 (Groves and Manning [ Reference Groves and ManningGM08 ]). Let $\Lambda$ be a graph. The combinatorial horoball based on $\Lambda$ , denoted by $\mathcal{H}\bigl(\Lambda\bigr)$ , is a graph constructed as follows:

(i) the vertex set is defined as $\mathcal{H}\bigl(\Lambda\bigr)^{(0)} \,\,:\!=\, \Lambda^{(0)} \times \mathbb{N}_0$ , where $\Lambda^{(0)}$ is the vertex set of $\Lambda$ ;

(ii) there are two kinds of edges in $\mathcal{H}\bigl(\Lambda\bigr)$ :

(i) for each $n\in \mathbb{N}_0$ and each $v \in \Lambda^{(0)}$ , there is a vertical edge in $\mathcal{H}\bigl(\Lambda\bigr)$ between (v, n) and $(v,n+1)$ ;

(ii) for each $n \in \mathbb{N}_0$ , and each pair of vertices (v, n) and (w, n), there is a horizontal edge between (v, n) and (w, n) if and only if $ 0 < d_{\Lambda}(v,w) \leq 2^n$ .
We denote by $\Lambda \times \{k\}$ the subgraph of $\mathcal{H}(\Lambda)$ spanned by the vertex set $\Lambda^{(0)} \times \{k\}$ .
Definition 2·2. A rough isometry is a quasiisometry with multiplicative constant 1.
Definition 2·3. Recall that, for each $k \in \mathbb{N}$ , the Rips graph $\text{Rips}_{k}(\Lambda$ ) of a graph $\Lambda$ is the graph with vertex set $\Lambda^{(0)}$ and edges consisting of pairs of vertices at distance at most k in $\Lambda$ .
Remark 2·4. Observe that the bijection $\Lambda^{(0)} \xrightarrow{\cong} \Lambda^{(0)} \times \{n\} \subset \mathcal{H}(\Lambda)^{(0)}$ given by $v \mapsto (v,n)$ extends to an isomorphism $\text{Rips}_{2^n}(\Lambda) \xrightarrow{\cong} \Lambda \times \{n\} \subset \mathcal{H}(\Lambda)$ . In particular, $\Lambda \times \{0\}$ is isomorphic to $\Lambda$ and, for each n, the subgraph $\Lambda \times \{n\}$ is roughly isometric to $\Lambda$ with the metric scaled by ${1}/{2^n}$ .
Definition 2·5 (Groves and Manning [ Reference Groves and ManningGM08 ]). Let $\Gamma$ be a graph and $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ be a family of subgraphs of $\Gamma$ . The augmented space $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is the graph obtained by attaching, for each $\alpha \in \mathscr{A}$ , the combinatorial horoball $\mathcal{H}\bigl(\Lambda_{\alpha}\bigr)$ to $\Gamma$ by identifying the subgraph $\Lambda_{\alpha} \subset \Gamma$ with the subgraph $\Lambda_{\alpha} \times \{0\} \subset \mathcal{H}\bigl(\Lambda_{\alpha}\bigr)$ along the isomorphism $\Lambda_{\alpha} \xrightarrow{\cong} \Lambda_{\alpha} \times \{0\}$ given by $v \mapsto (v,0)$ .
Definition 2·6. Let $\Gamma$ be a graph and $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ be a family of subgraphs of $\Gamma$ . Then $\Gamma$ is hyperbolic relative to $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ if the augmented space $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is $\delta$ hyperbolic for some $\delta$ . In that case, we call each $\Lambda_{\alpha} = \Lambda_{\alpha} \times \{0\}$ a parabolic subgraph of $\Gamma$ .
Remark 2·7. The above definition for graphs is motivated by the characterisation of relative hyperbolicity for groups by Groves and Manning (see Definition 2·11 below). Our definition is likely equivalent to metric notions of relative hyperbolicity as investigated in [ Reference SistoSis12 ], but we do not prove nor do we need such an equivalence for the purposes of this paper.
Definition 2·8. Let $\Gamma$ be a graph and $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ be a family of subgraphs of $\Gamma$ . The nrestricted augmentation $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is the subgraph of $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ spanned by the vertex set $\Gamma^{(0)} \sqcup \bigsqcup_{\alpha \in \mathscr{A}, k \in \{1, \ldots, n\}} \Lambda_{\alpha}^{(0)}\times \{k\}$ .
Similarly, the nrestricted horoball $\mathcal{H}_n\bigl(\Lambda\bigr)$ is the subgraph of the horoball $\mathcal{H}\bigl(\Lambda\bigr)$ spanned by the vertex set $\bigsqcup_{k \in \{1, \ldots, n\}} \Lambda \times \{k\}$ .
Remark 2·9. If a group G acts properly and cocompactly on $\Gamma$ and $(\Lambda_{\alpha})_{\alpha}$ is Ginvariant then G acts properly and cocompactly on $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ . Moreover, the embedding of $\Gamma$ in $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is Gequivariant and, for any $\alpha$ , the stabiliser of $\Lambda_{\alpha}\times \{n\}$ is equal to the stabiliser of $\Lambda_{\alpha}$ .
Remark 2·10. The graph $\Gamma$ is hyperbolic relative to $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ if and only if for each (any) $n \in \mathbb{N}_0$ , $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha \in \mathscr{A}}\bigr)$ is hyperbolic relative to $(\Lambda_{\alpha}\times \{n\})_{\alpha \in \mathscr{A}}$ . Thus, when we speak of the parabolics of $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha}\bigr)$ we will mean the top levels of the nrestricted horoballs $(\Lambda_{\alpha}\times \{n\})_{\alpha}$ .
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 ].
Definition 2·11. Let G be a finitely generated group and let $H_1, \ldots, H_k$ be a family of finitely generated subgroups of G. For $1 \leq i \leq k$ , let $S_i$ be a finite generating set for $H_i$ and let S be a finite generating set for G such that each $S_i \subset S$ . Denote by $\Gamma$ the Cayley graph $\text{Cayley}(G,S)$ and, for $1 \leq i \leq k$ , and $g \in G$ , denote by $g\Lambda_i$ the subgraph of $\Gamma$ with vertex set $gH_i$ and edges labelled by $gS_i$ . Then G is hyperbolic relative to $\{H_1, \ldots, H_k\}$ if $\Gamma$ is hyperbolic relative to $\{g\Lambda_i\}_{1 \leq i \leq k, g \in G}$ .
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 3·1 (See Bridson and Haefliger [ Reference Bridson and HaefligerBH99 , theorem III·H·1·13]). Let $\Gamma$ be a $\delta$ hyperbolic space and let $r > 8\delta + 1$ . Then there exists a constant $K = K(\delta,r)$ depending only on $\delta$ and r such that the following holds. If $\gamma$ is a path in $\Gamma$ and every subpath of length r of $\gamma$ is a geodesic then $\gamma$ is a $(2\delta,K)$ quasigeodesic.
Theorem 3·2 (See Bridson and Haefliger [ Reference Bridson and HaefligerBH99 , theorem III·H·1·7]). Let $\Gamma$ be a $\delta$ hyperbolic graph. Let $L > 0$ and $K \ge 0$ . Then there exists a constant $M = M(\delta,L, K)$ such that for any two (L, K)quasigeodesics $\beta_1$ and $\beta_2$ with the same endpoints, the images $\text{im}(\beta_1)$ and $\text{im}(\beta_2)$ are at Hausdorff distance at most M.
Lemma 3·3. Let $\mathcal{H}_n\bigl(\Lambda\bigr)$ be an nrestricted horoball. Let $v_1, v_2 \in \mathcal{H}_n\bigl(\Lambda\bigr)$ be given. The following hold:

(i) there exists a geodesic $\beta$ between $v_1,v_2$ whose image consists of at most two vertical segments and one horizontal segment. If the horizontal segment is not contained in $\Lambda \times \{n\}$ , then it is of length at most 3. Further, any geodesic between the two points is at Hausdorff distance at most 4 from $\text{im}(\beta)$ ;

(ii) if the horizontal segment of $\text{im}(\beta)$ is contained in $\Lambda\times \{K\}$ , then the image of any geodesic between $v_1$ and $v_2$ is disjoint from $\Lambda\times \{K^{\prime}\}$ for all $K^{\prime} > K$ .

(iii) Moreover, if k is the least number such that either $v_1$ or $v_2$ is contained in $\Lambda\times \{k\}$ , then the image of any geodesic between the points is contained in $\mathcal{H}_n\bigl(\Lambda\bigr) \setminus \mathcal{H}_{k1}\bigl(\Lambda\bigr)$ .
Lemma 3·3 is essentially a restatement 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) 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 $\Lambda\times \{mk\}$ to $\Lambda\times \{m\}$ . Similarly, the immediate successor of the horizontal segment is a descending segment. See Figure 2 for an illustration.

(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) Any geodesic path contains at most two maximal descending (respectively ascending) segments. See Figure 4.
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.
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.
Lemma 3·4 ([ Reference Groves and ManningGM08 , lemma 3·26]). Let $\Gamma$ be a graph that is hyperbolic relative to a family $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ of subgraphs. Let $\delta$ be the hyperbolicity constant of $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha\in \mathscr{A}}\bigr)$ . Then for any $k > \delta$ and any $\alpha \in \mathscr{A}$ , $\mathcal{H}\bigl(\Lambda_{\alpha}\bigr) \setminus \mathcal{H}_k\bigl(\Lambda_{\alpha}\bigr)$ is convex in $\mathcal{H}\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha\in \mathscr{A}}\bigr)$ .
Theorem 3·5. Let $\Gamma$ be a graph that is hyperbolic relative to a family $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ of subgraphs. Then, for n large enough, the parabolics (i.e. the top levels) of the restricted horoballs $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha\in \mathscr{A}}\bigr)$ are convex subgraphs.
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 nrestricted 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 nrestricted 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.
Note that by Lemma 3·3, each $\beta_i$ is a path which satisfies the following:

(i) $\text{im}(\beta_i)$ is not contained in any single nrestricted horoball and thus has length at least $2(nr) > 2r$ , and

(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 $\Lambda_{\alpha}\times \{nr\}$ (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 rlocal 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 rlocal geodesic since the rball 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 $\beta^{\prime}_{i1}\cdot\gamma^{\prime}_i\cdot\beta^{\prime}_i$ 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 rlocal geodesic. This proves the claim.
Thus by Lemma 3·1, $\gamma^{\prime}$ is a $(2\delta,K)$ quasigeodesic 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 $\mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr)\setminus \mathcal{H}_{n1}\bigl(\Lambda_{\alpha_0}\bigr)$ (again, by Lemma 3·4). This implies that $\gamma^{\prime}$ lies in $N_M(\mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr) \setminus \mathcal{H}_{n1}\bigl(\Lambda_{\alpha_0}\bigr)) \subset \mathcal{H}\bigl(\Lambda_{\alpha_0}\bigr) \setminus \mathcal{H}_{2r}\bigl(\Lambda_{\alpha_0}\bigr)$ .
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 $\Lambda_{\alpha_0}\times\{nr\}$ 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 nrestricted horoball, which is a contradiction.
Using Lemma 3·3 once again, we conclude that $\gamma \subset \Lambda_{\alpha_0}\times \{n\}$ .
Corollary 3·6. Let $\Gamma$ be a graph that is hyperbolic relative to a family $(\Lambda_{\alpha})_{\alpha \in \mathscr{A}}$ of subgraphs. Let n be such that the parabolics $(\Lambda_{\alpha}\times \{n\})_{\alpha}$ of $\mathcal{H}_n\bigl(\Gamma, (\Lambda_{\alpha})_{\alpha\in \mathscr{A}}\bigr)$ are convex subgraphs, as in Theorem 3·5. Then for each $\alpha \in \mathscr{A}$ , the subspace $\Lambda_{\alpha}^{(0)}\times \{0\}$ is roughly isometric to the subgraph $\Lambda_{\alpha}\times \{n\}$ .
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
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.
Theorem 1·3. 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 there is a connected, free cocompact Ggraph $\Gamma$ with subgraphs $(\Gamma_i)_i$ such that, for each i:

(i) $\Gamma_i$ is a Rips graph of $\text{Cayley}(H_i,S_i)$ ;

(ii) $H_i$ stabilises $\Gamma_i$ ;

(iii) the $H_i$ action on $\Gamma_i$ is free and cocompact; and

(iv) $\Gamma_i$ is convex in $\Gamma$ .
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 Gtranslates 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 ngons [ Reference HodaHod20 , definition 3·2].
Definition 4·1. Let $C_n$ denote the cycle graph of length n (i.e., a circle subdivided into n edges and n vertices) and let $C_n^{(0)}$ denote the vertex set of $C_n$ . A metric space X is strongly shortcut if there exists a $K > 1$ , an $n \in \mathbb{N}$ and an $M > 0$ such that there is no Kbilipschitz embedding of $(C_n^{(0)}, \lambda d_{C_n})$ in X with $\lambda \ge M$ .
Theorem 4·2 ([ Reference HodaHod20 , corollary 3·6]). A graph $\Gamma$ is strongly shortcut as a graph if and only if it is strongly shortcut as a metric space.
Our aim in this section is to prove the following.
Theorem 4·3. Let G be a finitely generated group that is hyperbolic relative to a family of strongly shortcut groups $(H_i)_i$ . Then G has a finite generating set S for which the $H_i$ are strongly shortcut metric subspaces of $\text{Cayley}(G,S)$ .
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 quasiisometry, up to scaling the metric on the Cayley graph. This arbitrary control on the multiplicative constant of the quasiisometry comes at the cost of having to choose larger and larger finite generating sets and accepting larger and larger additive quasiisometry 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
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
is a $(K, C_K)$ quasiisometry 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
is a quasiisometric 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 quasiisometry 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 quasiisometry $(H_i, t^{\prime} d_S) \to \bigl(H_i, {1}/{2^n}d_{S_i}\bigr)$ with multiplicative constant L. Scaling the domain and the codomain by $2^n$ , taking the quasiinverse 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 quasiisometry $(H_i, d_{S_i}) \hookrightarrow (G, 2^n t^{\prime} d_S)$ with multiplicative factor L.
Finally, we will need the next two theorems about strongly shortcut spaces and groups.
Theorem 4·6 ([ Reference HodaHod20 , proposition 3·4]). Let X be a strongly shortcut metric space. Then there exists an $L_X > 1$ such that whenever Y is a metric space and $C > 0$ and $f \colon Y \to X$ is an $(L_X,C)$ quasiisometry up to scaling, then Y is also strongly shortcut.
Theorem 4·7 ([ Reference HodaHod20 , theorem C]). A group G is strongly shortcut if and only if G has a finite generating set S for which $\text{Cayley}(G,S)$ is strongly shortcut.
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 quasiisometric 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 quasiisometric 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 treegradedness 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 ].
Definition 5·1. A nonprincipal ultrafilter $\omega$ over $\mathbb{N}$ is a set of subsets of $\mathbb{N}$ satisfying the following properties:

(i) for each $A \subseteq \mathbb{N}$ , either $A \in \omega$ or $\mathbb{N} \setminus A \in \omega$ , but not both;

(ii) no finite subset of $\mathbb{N}$ is in $\omega$ ;

(iii) if $A,B \in \omega$ , then $A \cap B \in \omega$ ;

(iv) if $A \in \omega$ and $A \subseteq B$ , then $B \in \omega$ .
The existence of nonprincipal ultrafilters is a consequence of Zorn’s Lemma (see [ Reference Druţu and KapovichDK18 , lemma 10·18] for instance).
Definition 5·2. Let $\omega$ be a nonprincipal ultrafilter. Let $(x_n)_n$ be a sequence of points in a topological space X. An element $x \in X$ is an $\omega$ limit of $(x_n)_n$ , denoted $\lim_{\omega} x_n$ , if for every open set $U \ni x$ , the set $A_U = \{n \in \mathbb{N}  x_n \in U\}$ is contained in $\omega$ .
Remark 5·3. If X is a Hausdorff space, then an $\omega$ limit is unique whenever it exists. If X is compact, then for every sequence, an $\omega$ limit exists.
Let (X, d) be a metric space and let $\omega$ be a nonprincipal 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 $X_B^{\mathbb{N}}((r_n)_n,(p_n)_n) \,:\!=\, \{(x_n)_n \in X^{\mathbb{N}}  d_{\infty}((x_n)_n,(p_n)_n) < \infty\}$ .
Remark 5·4. Note that $(X_B^{\mathbb{N}}((r_n)_n,(p_n)_n),d_{\infty})$ is a pseudometric space.
Definition 5·5. The asymptotic cone $\text{Cone}_{\omega}(X,(r_n)_n,(p_n)_n)$ of X is the quotient of $X_B^{\mathbb{N}}((r_n)_n,(p_n)_n)$ identifying $(x_n)_n$ and $(y_n)_n$ whenever $d_{\infty}((x_n)_n,(y_n)_n) = 0$ . We let $[x_n]$ denote the point of $\text{Cone}_{\omega}(X,(r_n)_n,(p_n)_n)$ represented by $(x_n)_n$ .
Remark 5·6. For a group G equipped with a left invariant metric, any asymptotic cone $\text{Cone}_{\omega}(G,(r_n)_n,(p_n)_n)$ is isometric to $\text{Cone}_{\omega}(G,(r_n)_n,(1)_n)$ , where $(1)_n$ is the constant basepoint sequence at the identity. Thus in this case we will simply write $\text{Cone}_{\omega}(G,(r_n)_n)$ .
Definition 5·7. (Drutu and Sapir [ Reference Druţu and SapirDS05 ]) A complete geodesic metric space X is a tree graded space with respect to a collection of closed geodesic subspaces, called pieces, if the following two properties are satisfied:

(i) any two distinct pieces intersect in at most a single point, and

(ii) every nontrivial simple geodesic triangle (i.e., the concatenation of the three geodesics is a simple loop) in X is contained in a piece.
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
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
and $d_{\infty}$ is a leftinvariant metric with respect to this group structure. Thus $\lim_{\omega} (g_n H_i)_n$ is isometric to
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 $C\mathbb{Z}$ .
Theorem 5·10. ([ Reference HodaHod20 , theorem 3·7]). A metric space X is strongly shortcut if and only if no asymptotic cone of X contains an isometric copy of the Riemannian circle of unit length.
We are now ready to prove our main result, which we first recall:
Theorem 1·2. Let G be a finitely generated group that is hyperbolic relative to strongly shortcut groups. Then G is strongly shortcut.
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.
Acknowledgements
This work was supported by Polish Narodowe Centrum Nauki UMO2017/25/B/ST1/01335 as well as by the grant 346300 for IMPAN from the Simons Foundation and the matching 20152019 Polish MNiSW fund. The collaboration that led to this article was initiated at the 2019 Simons Semester in Geometric and Analytic Group Theory in Warsaw.
The first named author was supported by the ERC grant GroIsRan and an NSERC Postdoctoral Fellowship. The second named author was supported by CEFIPRA grant number 58011, “Interactions between dynamical systems, geometry and number theory” at Tata Institute of Fundamental Research and by grant number ISF 1226/19 at the Technion.
We thank the anonymous referee for helpful inputs which improved the exposition of this paper.