Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut

Abstract We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.

1. Introduction 2 2. Relative hyperbolicity à la Groves and Manning 3 3. Horoballs and convexity of parabolics 5 4. A Cayley graph with strongly shortcut parabolics 10 5. Asymptotic cones and the proof of the main result 1. Introduction Strongly shortcut graphs and groups were introduced by the first named author [Hod18] who later generalized the strong shortcut property to rough geodesic metric spaces [Hod20].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 Gromov-hyperbolic spaces [Hod18], asymptotically CAT(0) spaces [Hod20], hierarchically hyperbolic spaces, coarse Helly metric spaces of uniformly bounded geometry [HHP20], 1skeletons of finite dimensional CAT(0) cube complexes (i.e.median graphs), 1skeletons of quadric complexes (i.e.hereditary modular graphs), 1-skeletons of systolic complexes (i.e.bridged graphs), standard Cayley graphs of Coxeter groups [Hod18] and all of the Thurston geometries except Sol [HP,Kar11].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 [Hod18,Hod20].
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 Γ is strongly shortcut if, for some K > 1 there is a bound on the lengths of the K-bilipschitz combinatorial cycles in Γ.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 G-action 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 [Gro87], asymptotically CAT(0) groups [Kar11] (e.g.CAT(0) groups [BH99]), hierarchically hyperbolic groups [BHS17, BHS19] (e.g.mapping class groups of surfaces [MM99, MM00]), coarse Helly groups [CCG + 20] (e.g.Artin groups of FC-type, weak Garside groups [HO19]), the discrete Heisenberg group [HP], systolic groups (e.g.finitely presented C(6) small cancellation groups [Wis03]) and quadric groups (e.g.C(4)-T (4) small cancellation groups) [Hod17].
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 t be a maximal cyclic subgroup generated by a loxodromic element t of the Bass-Serre tree of G. Then the amalgamated free product G * t G is hyperbolic relative to discrete Heisenberg subgroups by Dahmani [Dah03] and thus is strongly shortcut by Theorem 1.2 and [HP].
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 characterizes strongly shortcut groups as those whose asymptotic cones have no isometrically embedded circles ([Hod20, Theorem 3.7]), while a result of Osin and Sapir [DS05, Theorem A.1] guarantees that asymptotic cones of relatively hyperbolic groups are tree-graded.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 [GM08] 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 G-graph Γ with subgraphs (Γ i ) i such that, for each i, (1 the H i action on Γ i is free and cocompact, and (4) Γ i is convex in Γ.
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 characterization 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.
Acknowledgements.This work was supported by Polish Narodowe Centrum Nauki UMO-2017/25/B/ST1/01335 as well as by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 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 CE-FIPRA grant number 5801-1, "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.

Relative hyperbolicity à la Groves and Manning
Definition 2.1 (Groves and Manning [GM08]).Let Λ be a graph.The combinatorial horoball based on Λ, denoted by H Λ , is a graph constructed as follows: • The vertex set is defined as H Λ (0) := Λ (0) × N 0 , where Λ (0) is the vertex set of Λ. • There are two kinds of edges in H Λ : (1) For each n ∈ N 0 and each v ∈ Λ (0) , there is a vertical edge in H Λ between (v, n) and (v, n + 1).(2) For each n ∈ 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 We denote by Λ × {k} the subgraph of H(Λ) spanned by the vertex set Λ (0) × {k}.
Definition 2.3.Recall that, for each k ∈ N, the Rips graph Rips k (Λ) of a graph Λ is the graph with vertex set Λ (0) and edges consisting of pairs of vertices at distance at most k in Λ.
Remark 2.7.The above definition for graphs is motivated by the characterization 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 [Sis12], but we do not prove nor do we need such an equivalence for the purposes of this paper.
Definition 2.8.Let Γ be a graph and (Λ α ) α∈A be a family of subgraphs of Γ.The n-restricted augmentation Remark 2.9.If a group G acts properly and cocompactly on Γ and (Λ α ) α is Ginvariant then G acts properly and cocompactly on H n Γ, (Λ α ) α∈A .Moreover, the embedding of Γ in H n Γ, (Λ α ) α∈A is G-equivariant and, for any α, the stabilizer of Λ α × {n} is equal to the stabilizer of Λ α .
The following definition is due to Groves and Manning, who prove that it is equivalent to strong relative hyperbolicity [Far98,Bow12].We refer the reader to [GM08, 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 [Hru10].
Definition 2.11.Let G be a finitely generated group and let H 1 , . . ., H k be a family of finitely generated subgroups of G.For 1 ≤ i ≤ 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 ⊂ S. Denote by Γ the Cayley graph Cayley(G, S) and, for 1 ≤ i ≤ k, and g ∈ G, denote by gΛ i the subgraph of Γ with vertex set gH i and edges labelled by

Horoballs and convexity of parabolics
It is well-known that given a relatively hyperbolic group, its parabolic subgroups are quasiconvex [DS05, 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 [BH99, Theorem III.H.1.13]).Let Γ be a δ-hyperbolic space and let r > 8δ + 1.Then there exists a constant K = K(δ, r) depending only on δ and r such that the following holds.If γ is a path in Γ and every subpath of length r of γ is a geodesic then γ is a (2δ, K)-quasi-geodesic.

The following hold:
(1) There exists a geodesic β 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 Λ × {n}, then it is of length at most 3. Further, any geodesic between the two points is at Hausdorff distance at most 4 from im(β).(2) If the horizontal segment of im(β) is contained in Λ × {K}, then the image of any geodesic between v 1 and v 2 is disjoint from Λ × {K ′ } for all K ′ > K.
(3) Moreover, if k is the least number such that either v 1 or v 2 is contained in Λ × {k}, then the image of any geodesic between the points is contained in Lemma 3.3 is essentially a re-statement of Lemma 3.10 of [GM08] 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 γ 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.
Figure 2. The red path between (x, m) and (y, m + 1) on the left is longer than the red path on the right.
(1) Assume that a geodesic path contains a horizontal segment in Λ × {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 Λ × {m − k} to Λ × {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 Λ × {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 γ be a geodesic between the points v 1 and v 2 in the statement.By the above observations, if γ contains a horizontal segment of length at least two at some Λ × {m}, then im(γ) is disjoint from Λ × {m ′ } for all m < m ′ ≤ n.Thus, any horizontal segment in γ is either of length one, or is contained in the maximum level Λ × {max} that intersects im(γ) nontrivially.
In fact, it can be verified that apart from the horizontal segment at Λ × {max}, the image of γ can have at most one more horizontal edge.
Another consequence of the above observations is that if γ contains a horizontal segment of length at least 6, then this segment has to be contained in Λ × {n}, see Figure 5.
Assume that the horizontal edge not at Λ × {max} is an edge between (x, m) and (y, m) and is followed by an ascending segment from (y, m) to (y, max) ∈ Λ×{max}.Let γ ′ be the geodesic obtained from γ 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 γ ′ contains 4 or 5 edges, then let β be the geodesic obtained by replacing this horizontal segment by an ascending edge, a horizontal segment in Λ × {max +1} and a descending edge back to Λ × {max}, similar to the procedure in Figure 5.We leave it as an exercise to verify that β is as required.
Before stating the main result of this section, we recall a convexity result from [GM08] which will be used in the proof.
Lemma 3.4 (Lemma 3.26, [GM08]).Let Γ be a graph that is hyperbolic relative to a family (Λ α ) α∈A of subgraphs.Let δ be the hyperbolicity constant of H Γ, (Λ α ) α∈A .Then for any k > δ and any Theorem 3.5.Let Γ be a graph that is hyperbolic relative to a family (Λ α ) α∈A of subgraphs.Then, for n large enough, the parabolics (i.e. the top levels) of the restricted horoballs H n Γ, (Λ α ) α∈A are convex subgraphs.
Proof.Let δ be the hyperbolicity constant of H Γ, (Λ α ) α∈A .Let r = ⌈8δ + 2⌉ and n ≥ 2r + M (δ, 2δ, K), where K is the constant from Lemma 3.1 and M is the constant from Theorem 3.2.Fix α 0 ∈ A and points x, y ∈ Λ α0 × {n}.Let γ : P → H n Γ, (Λ α ) α∈A be a geodesic (in H n Γ, (Λ α ) α∈A ) between x and y.Since each n-restricted horoball in H n Γ, (Λ α ) α∈A is a full subgraph, every subpath of γ whose image lies in an n-restricted horoball is a geodesic in that horoball.We will therefore assume that each such geodesic subpath of γ is of the form given by Lemma 3.3.
Denote by U ⊂ H n Γ, (Λ α ) α∈A the set α∈A N r Λ α × {n} .The path γ is a concatenation where each γ i is a path with image in U and each β 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 β i is a path which satisfies the following: • im(β i ) is not contained in any single n-restricted horoball and thus has length at least 2(n − r) > 2r, and • for any α ∈ A , im(β i ) ∩ H n Λ α is a union of components, where each component is either a vertical segment between Λ α × {0} and Λ α × {n − r} (e.g., im(β 2 ) ∩ H n Λ α2 in Figure 6), or the image of a geodesic between points of Λ α × {0} (e.g., im(β 1 ) ∩ H n Λ α1 in Figure 6).In the latter case, we note that this component is disjoint from the image of any γ j .Let ι : H n Γ, (Λ α ) α∈A ֒→ H Γ, (Λ α ) α∈A denote the inclusion map.For each i, let γ ′ i be a geodesic path in H Γ, (Λ α ) α∈A between the endpoints of ι

Proof of claim. Each γ ′
i is a geodesic, and therefore a local geodesic.Each β ′ i is an r-local geodesic since the r-ball around any point in im( As observed above, the image of every subpath of β ′ i that lies in a horoball is either a vertical segment or it does not meet any γ ′ j .This implies that any subpath of Figure 6.The path γ is a concatenation of paths γ i (in orange) and β j (in blue).
Corollary 3.6.Let Γ be a graph that is hyperbolic relative to a family (Λ α ) α∈A of subgraphs.Let n be such that the parabolics (Λ α × {n}) α of H n Γ, (Λ α ) α∈A are convex subgraphs, as in Theorem 3.5.Then for each α ∈ A , the subspace Proof.Let (x, 0), (y, 0) ∈ Λ α × {0} be vertices at the bottom level of the combinatorial horoball based on Λ α in Γ and let (x, n), (y, n) ∈ Λ α × {n} be the corresponding vertices at the nth level.We have 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 G-graph Γ with subgraphs (Γ i ) i such that, for each i, (1) Γ i is a Rips graph of Cayley(H i , S i ), (2) H i stabilizes Γ i , (3) the H i action on Γ i is free and cocompact, and (4) Γ i is convex in Γ.
Proof.Let S be a finite generating set of G containing each of the S i .Let Γ be the Cayley graph of G with respect to S. Then the Cayley graphs Γ ′ i = Cayley(H i , S i ) are subgraphs of Γ and G is hyperbolic relative to the family (gΓ ′ i ) g,i of G-translates of these subgraphs.By Theorem 3.5, there is an n for which the parabolics of H n Γ, (gΓ ′ i ) g,i are convex.For each i, let Γ i be the parabolic in the restricted horoball with base Γ ′ i .Then H n and the Γ i satisfy all the required conditions.

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 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.denote the vertex set of C n .A metric space X is strongly shortcut if there exists a K > 1, an n ∈ N and an M > 0 such that there is no K-bilipschitz embedding of (C (0) n , λd Cn ) in X with λ ≥ M .Theorem 4.2 ([Hod20, Corollary 3.6]).A graph Γ is strongly shortcut as a graph if and only if it is strongly shortcut as a metric space.
Our goal 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 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 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 [Hod20, 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 ∈ X.For t > 0 let S t be the finite set defined by S t = g ∈ G : d(x 0 , gx 0 ) ≤ t and consider the word metric d St defined by S t .(For those t where S t does not generate G, we allow d St to take the value ∞).Let K t be the infimum of all K > 1 for which 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 Si ) ֒→ (G, td S ) is a quasi-isometric embedding with multiplicative constant L, where d S and the d Si are the word metrics.
Proof.Let S ′ ⊇ i S i be a finite generating set for G. Let Γ ′ = Cayley(G, S ′ ) and let Λ g,i = g Cayley(H i , S i ).By Theorem 3.5, for some n, the top level subgraphs Λ g,i × {n} of the restricted horoballs of H n = H n Γ ′ , {Λ g,i } g,i are convex.Moreover, by Remark 2.9, the group G acts properly and cocompactly on H n .
By Corollary 3.6 and Remark 2.4, there is a rough isometry (Λ e,i × {0}, d Hn ) → H i , 1 2 n d Si .By Theorem 4.4, there is a generating set S for G and a scaling factor t ′ > 0 such that the inclusion (G, t ′ d S ) ֒→ 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 Λ 2 n d Si 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 ′ d S ) ֒→ (G, 2 n t ′ d S ) we obtain a quasi-isometry (H i , d Si ) ֒→ (G, 2 n t ′ d S ) with multiplicative factor L.
Finally, we will need the next two theorems about strongly shortcut spaces and groups.
Theorem 4.6 ([Hod20, 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 : Y → X is an (L X , C)-quasi-isometry up to scaling, then Y is also 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 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 Si ) 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 , td S ) is quasi-isometric to (H i , d Si ) with multiplicative constant L = min i L i .Thus each (H i , d S ) is strongly shortcut.

Figure 3 .Figure 4 .Figure 5 .
Figure 3.The red path between (x, m) and (y, m) on the bottom panel is shorter than the one on the top panel.
The following definition appears in earlier work of the first named author under the name nonapproximability of ngons [Hod20, 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 (0) n × {0} and so the composition of the restriction (H i , t ′ d S ) ֒→ (Λ (0) e,i × {0}, d Hn ) and the rough isometry (Λ (0) e,i × {0}, d Hn ) → H i , 1 2 n d Si gives us a quasi-isometry (H i , t ′ d S ) → H i , 1 Theorem 4.7 ([Hod20, Theorem C]).A group G is strongly shortcut if and only if G has a finite generating set S for which Cayley(G, S) is strongly shortcut.