On Subdirect Products of Type $ FP N $ of Limit Groups Over Droms Raags

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Wilton proved that finitely presentable subgroups of finitely generated residually free groups are separable and that the subgroups of type FP8 are virtual retracts. Recently, in [21], J. Lopez de Gamiz Zearra has shown that in a finitely generated residually Droms RAAG every finitely presented subgroup is subgroup separable and that every subgroup of type FP8pQq is a virtual retract. Though limit groups over Droms RAAGs share many properties with limit groups over free groups, there are properties like commutative transitivity and the Howson property that do not hold for a general Droms RAAG. For example, by [15], the RAAGs that satisfy the Howson property are precisely the free products of abelian groups, so in particular, are limit groups over free groups. In [19] D. Kochloukova showed that limit groups over free groups are free-by-(torsion-free nilpotent). Our first result is a generalization of this fact for limit groups over Droms RAAGs. The rest of the results in this paper will be heavily dependent on this fact.

Proposition A Every limit group over Droms RAAGs is free-by-(torsion-free nilpotent).

Let us recall some definitions. Let S be a subgroup of a direct product G1 ˆ¨¨¨ˆ Gn. We say that S is full if S X Gi , 1 for all i P t1,..., nu and S is a subdirect product if pipSq“ Gi for all i P t1,..., nu, where pi is the projection map G1 ˆ¨¨¨ˆ Gn ÞÑ Gi. G. Baumslag, A. Miasnikov, V.Remeslennikov proved in [4] that every finitely generated residually Droms RAAG is a subgroup of a direct product of finitely many limit groups over Droms RAAGs. The study of subdirect products of free groups was initiated by G. Baumslag and J. Roseblade in [5]. In a sequence of papers that culminated in [6] and [7], M. Bridson, J. Howie, C. Miller and H. Short studied subgroups of the direct product of limit groups over free groups. One of the results is that if Γ1,..., Γn are non-abelian limit groups over free groups and S is a full subdirect product of Γ1 ˆ¨¨¨ˆ Γn, then S is of type FP2pQq if and only if pi,jpSq is of finite index in Γi ˆ Γj for all 1 ď i ă j ď n, where pi,j denotes the projection map S ÞÑ Γi ˆ Γj. These results were generalized by J. Lopez de Gamiz Zearra in [21] to the class of limit groups over Droms RAAGs (see Section 2.2). In addition, in [19] D. Kochloukova generalised the previous result concerning limit groups over free groups by showing that if S is of type FPs, then it virtually surjects onto s coordinates. However, the converse is still an open problem.

We recall that a group S is of homological type FPspRq (or it is FPs over R) for a commutative, associative ring R with identity element if the trivial RG-module R has a projective resolution with all modules ﬁnitely generated in dimensions up to s. A group is of type FPs if it is FPs over Z. The goal of Section 4 is to extend the above results to the class of limit groups over Droms RAAGs.

Theorem B Let G1,..., Gm be limit groups over Droms RAAGs such that each Gi has trivial center and let S ă G ≔ G1 ˆ¨¨¨ˆ Gm be a finitely generated full subdirect product. Suppose further that, for some fixed number s P t2,..., mu, for every subgroup S0 of finite index in S the homology group HipS0, Qq is finite dimensional (over Q) for all i ď s. Then for every canonical projection

pj1,...,js : S ÞÑ Gj1 ˆ¨¨¨ˆ Gjs the index of pj1,...,js pSq in Gj1 ˆ ... ˆ Gjs is ﬁnite. 2 Q In particular, if S is of type FPs over , then for every canonical projection pj1,...,js the index of pj1,...,js pSq in

Gj1 ˆ¨¨¨ˆ Gjs is ﬁnite.

In Section 5 we discuss the growth of homology groups and the volume gradients of limit groups over Droms RAAGs and of ﬁnitely presented residually Droms RAAGs. The growth of homology groups is measured by the limit

lim dimK HipBn, Kq{rG: Bns, nÑ8 whenever this limit exits. Here K is a field and pBnq is an exhausting normal chain of G, that is, a chain pBnq of normal subgroups of finite index such that Bn`1 Ď Bn and n Bn “ 1. As every residually finite group has an exhausting normal chain, the same holds forŞ limit groups over coherent RAAGs. Thanks to the approximation theorem of W. Lück we know that if K has characteristic 0, G is residually finite, finitely presented and of type FPm, then the limit exists for 2 i ă m and equals the L -Betti number βipGq. In particular, in this case the limit is independent of the exhausting normal chain.

Given a group G of homotopical type Fm, volmpGq was defined in [8] as the least number of m-cells among all classifying spaces KpG, 1q with finite m-skeleton. For instance, vol1pGq equals dpGq, where dpGq is the minimal number of generators of G. The most studied volume gradient is the 1-dimensional one. That is, the rank gradient of G with respect to pBnq, which is defined to be limn dpBnq{rG: Bns and it is denoted by RGpG, pBnqq. Its study was initiated by M. Lackenby in [20] and a combinatorial approach to the rank gradient was developed by M. Abert, N. Nikolov and A. Jaikin-Zapirain in [1]. In general, the m-dimensional volume gradient of G with respect to pBnq is

lim volmpBnq{rG: Bns. nÑ8 In [8] M. Bridson and D. Kochloukova calculated the above volume and homology type gradients for limit groups over free groups and in the case of residually free groups they found particular filtrations where the homology growth and the rank gradient can be calculated. The results about homology growth in [8] and in this paper are independentof the characteristic of the field K. In Section 5 we compute the homology growth and the volume gradients for a more general setting, namely, for limit groups over coherent RAAGs (see Section 5.2 for the definition). Note that this class of groups is more general than the class of limit groups over Droms RAAGs. We use the approach from [8] and we show that limit groups over coherent RAAGs are slow above dimension 1, hence are K-slow above dimension 1. This implies the following results.

Theorem C Let G be a limit group over a coherent RAAG and let pBnq be an exhausting normal chain in G. Then,

dpBnq (1) the rank gradient RGpG, pB qq “ lim Ñ8 “´χpGq. n n rG: Bns

defpBnq (2) the deﬁciency gradient DGpG, pB qq “ lim Ñ8 “ χpGq. n n rG: Bns

volkpBnq (3) the k-dimensional volume gradient lim Ñ8 “ 0 for k ě 2. n rG: Bns

Theorem D Let K be a ﬁeld, G a limit group over a coherent RAAG and pBnq an exhausting normal chain in G. Then,

dimK H1pBn,Kq (1) lim Ñ8 “´χpGq. n rG: Bns

dimK HjpBn,Kq (2) lim Ñ8 “ 0 for all j ě 2. n rG: Bns 3 Using results from [8] we also compute the homology growth up to dimension m and the volume gradients in low dimensions of residually Droms RAAGs of type FPm for m ě 2. We cannot apply the same method to the class of residually coherent RAAGs since Theorem B is a key result in this method and in [13] it is proved that this no longer holds for coherent RAAGs.

Theorem E Let m ě 2 be an integer, let G be a residually Droms RAAG of type FPm, and let ρ be the largest integer such that G contains a direct product of ρ non-abelian free groups. Then, there exists an exhausting sequence pBnq in G so that for all ﬁelds K,

(1) if G is not of type FP8, then

dim HipBn, Kq lim “ 0 for all 0 ď i ď m; nÑ8 rG: Bns

(2) if G is of type FP8 then for all j ě 1,

dim HρpBn, Kq dim HjpBn, Kq lim “ p´1qρχpGq, lim “ 0 for all j , ρ. nÑ8 rG: Bns nÑ8 rG: Bns

Remark The exhausting sequence pBnq will be constructed to be an exhausting normal chain in a subgroup of ﬁnite index in G.

The following result is a low dimensional homotopical version of Theorem E.

Theorem F Every group G that is a ﬁnitely presented residually Droms RAAGs but is not a limit group over Droms RAAGs admits an exhausting normal chain pBnq in G with respect to which the rank gradient

dpBnq RGpG, pBnqq “ lim “ 0. nÑ8 rG: Bns

Furthermore, if G is of type FP3 but it is not commesurable with a product of two limit groups over Droms RAAGs, pBnq can be chosen so that the deﬁciency gradient DGpG, pBnqq “ 0.

Acknowledgements We are thankful to Montserrat Casals-Ruiz for helpful talks during the prepa- ration of this manuscript. D. Kochloukova was partially supported by the CNPq grant 301779/2017- 1 and by the FAPESP grant 2018/23690-6. J. Lopez de Gamiz Zearra was supported by the Basque Government grant IT974-16 and the Spanish Government grant MTM2017-86802-P.

2. Preliminaries 2.1. Right-angled Artin groups. Let us recall the deﬁnition of a right-angled Artin group. Given a simplicial graph X with vertex set VpXq, the corresponding right-angled Artin group (RAAG), denoted by GX, is given by the following presentation: GX “xVpXq | xy “ yx ðñ x and y are adjacenty.

Subgroups of RAAGs can be wild, in particular, not all subgroups of RAAGs are themselves RAAGs. RAAGs that have all its ﬁnitely generated subgroups again of this type are known as Droms RAAGs.

Deﬁnition 2.1. Let G be a class of groups. The Z˚-closure of G, denoted by Z ˚ pGq, is the union of classes pZ ˚ pGqqk deﬁned as follows. At level 0, the class pZ ˚ pGqq0 equals G. A group G lies in pZ ˚ pGqqk if and only if m G » Z ˆ pG1 ˚¨¨¨˚ Gnq, 4 where m P N Y t0u and the group Gi lies in pZ ˚ pGqqk´1 for all i P t1,..., nu.

The level of G, denoted by lpGq, is the smallest k for which G belongs to pZ ˚ pGqqk.

In this terminology, Droms showed in [17] that the class of Droms RAAGs is the Z ˚ćlosure of Z. Analogously, it is the Z ˚ćlosure of the class of finitely generated free groups. We use the latter when we fix the length of a Droms RAAG.

Example 2.2. If G is a Droms RAAG such that lpGq “ 0, then G is a finite rank free group. If lpGq“ 1, then G » Zm ˆ F, for some m ě 1, F free of finite rank and G is not Z. If lpGq“ 2, then Zm Zn1 Znl G » ˆ p ˆ Fk1 q˚¨¨¨˚p ˆ Fkl q , N ` ˘ for some m, ni, ki P Y t0u, i ni ě 1, l ě 2 and for i P t1,..., lu, Fki free of finite rank ki. ř 2.2. Limit groups over Droms RAAGs. If G is a group, a limit group over G is a group H that is finitely generated and fully residually G; that is, for any finite set of non-trivial elements S Ď H there is a homomorphism ϕ: H ÞÑ G which is injective on S. Note that when G is a Droms RAAG this definitioncoincides with the definition of a finitely generated group H0 that is a fully residually Droms RAAG from the introduction. Indeed, if for any finite set of non-trivial elements S Ď H0 there is a homomorphism ϕ0 : H0 ÞÑ GS which is non-trivial on all s P S, where GS is a Droms RAAG, then by subtituting GS (if necessary) with impϕ0q we can assume that dpGSqď dpH0q. But since there are only finitely many Droms RAAGs with at most dpH0q generators, say D1,..., Dt, we conclude that GS P tD1,..., Dtu and GS is a subgroup of the fixed group G ≔ D1 ˚¨¨¨˚ Dt. By [4], groups that are residually Droms RAAGs are precisely finitely generated subgroups of a direct product of limit groups over Droms RAAGs. In this section we state the basic properties of residually Droms RAAGs and limit groups over Droms RAAGs thatweneedinlatersections. Limit groups over Droms RAAGs admit a hierarchical structure that comes from the fact that Droms RAAGs are the direct product of a free abelian group (possibly trivial) and a free product, so we can use the work of M. Casals-Ruiz and I. Kazachkov on limit groups over free products (see [12]) to obtain that hierarchy.

Proposition 2.3. [7, Proposition 2.1] Let G be a Droms RAAG such that lpGq “ 0 (that is, a ﬁnitely generated free group) and let Γ be a limit group over G of height hpΓq.

If hpΓq “ 0, then Γ equals M1 ˚¨¨¨˚ Mj and M1,..., Mj are free abelian groups or surface groups with Euler characteristic at most ´2. If hpΓq ě 1, then Γ acts cocompactly on a tree T where the edge stabilizers are trivial or inﬁnite cyclic and the vertex groups are limit groups over G of height at most hpΓq´ 1. Moreover, at least one of the vertex groups is nonabelian.

Proposition 2.4. [12, Theorem 8.2] Let G be a Droms RAAG with lpGq ě 1 so that G is of the form m Z ˆ pG1 ˚¨¨¨˚ Gnq and lpGiqď lpGq´ 1 for i P t1,..., nu and let Γ be a limit group over G. Then, Γ is l Z ˆ Λ where Λ is a limit group over G1 ˚¨¨¨˚ Gn and if m “ 0, then l “ 0. If hpΛq“ 0, then

Λ “ A1 ˚¨¨¨˚ Aj, where for each t P t1,..., ju,At is a limit group over Gi for some i P t1,..., nu. 5 If hpΛqě 1, then Λ acts cocompactly on a tree T where the edge stabilizers are trivial or inﬁnite cyclic and the vertex groups are limit groups over G1 ˚¨¨¨˚ Gn of height at most hpΛq´ 1. Moreover, at least one of the vertex groups has trivial center.

We observe that the notion of a height in limit groups over Droms RAAGs given in Proposition 2.4 is a natural generalization of the notion of a height in limit groups over free groups given in Proposition 2.3 except that, for historical reasons, the deﬁnition of limit groups over free groups of height 0 includes more than free products of free abelian groups and allows the free factors to be surface groups with Euler characteristic at most ´2. In [21], following the paper [7], the results concerning subgroups of direct products of limit groups over free groups were generalised to the case of limit groups over Droms RAAGs.

Theorem 2.5. [21, Theorem 3.1] If Γ1,..., Γn are limit groups over Droms RAAGs and S is a subgroup of Γ1 ˆ¨¨¨ˆ Γn of type FPnpQq, then S is virtually a direct product of limit groups over Droms RAAGs.

Theorem 2.6. [21, Theorem 8.1] Let Γ1,..., Γn be limit groups over Droms RAAGs such that each Γi has trivial center and let S ă Γ1 ˆ¨¨¨ˆ Γn be a ﬁnitely generated full subdirect product. Then either: (1) S is of ﬁnite index; or

(2) S is of infinite index and has a finite index subgroup S0 ă S such that HjpS0, Qq has infinite dimension for some j ď n.

Another result that will be used later is associated to finitely presented residually Droms RAAGs. In order to state the result, we introduce the following definition: an embedding S ãÑ Γ0 ˆ¨¨¨ˆ Γn of a finitely generated group S that is a residually Droms RAAG as a full subdirect product of limit groups over Droms RAAGs is neat if Γ0 is abelian (possibly trivial), S X Γ0 is of finite index in Γ0 and Γi has trivial center for i P t1,..., nu.

Theorem 2.7. [21, Theorem 10.1] Let S be a ﬁnitely generated group that is a residually Droms RAAG. The following are equivalent: (1) S is ﬁnitely presentable;

(2) S is of type FP2pQq;

(3) dim H2pS0, Qq is ﬁnite for all subgroups S0 ă S of ﬁnite index;

(4) there exists a neat embedding S ãÑ Γ0 ˆ¨¨¨ˆ Γn into a product of limit groups over Droms RAAGs such that the image of S under the projection to Γi ˆ Γj has ﬁnite index for 0 ď i ă j ď n; (5) for every neat embedding S ãÑ Γ0 ˆ¨¨¨ˆ Γn into a product of limit groups over Droms RAAGs the image of S under the projection to Γi ˆ Γj has ﬁnite index for 0 ď i ă j ď n.

3. Limit groups over Droms RAAGs are free-by-(torsion-free nilpotent)

Deﬁnition 3.1. A graph is called triangulated if it contains no induced copy of Cn, for n ě 4, where Cn is the circle with n vertices.

Theorem 3.2. [18, Theorem 2] If G is the RAAG corresponding to the graph X, the commutator subgroup G1 is free if and only if X is triangulated.

Recall that Droms RAAGs can be described as the RAAGs where the deﬁning graph does not contain induced squares or straight line paths with 3 edges (see [17]). In particular, if G is a Droms RAAG, then G1 is free. 6 Lemma 3.3. Droms RAAGs are free-by-(free abelian).

Proof. Let G be a Droms RAAG corresponding to the graph X. Then, G1 is free, the abelianization is Zn where n is the number of vertices in the graph X and there is a short exact sequence

1 G1 G Zn 1.

A ﬁltration tGiuiě1 of normal subgroups of a group G has Property (1) if G{Gi is torsion-free and iě1 Gi “ 1, and it has Property (2) if iě1 Gi “ 1 and for each ﬁnitely generated abelian subgroup ŞM of G there is i “ ipMq such that Gi XŞM “ 1.

Lemma 3.4. Let G be a group and tGiuiě1 be a ﬁltration of normal subgroups of G. If tGiuiě1 has Property (1), it also has Property (2).

Proof. Let M be a finitely generated abelian subgroup of G. Since iě1 Gi “ 1, there is i such that Gi X M is not M. The group M{pM X Giq embeds in G{Gi, so it is non-trivialŞ and torsion-free, that is, a finite rank free abelian group. As M has finite Hirsch length, we deduce that M X Gj is trivial for sufficiently large j.

Lemma 3.5. Let H be a group, let G be Zm ˆ H for some m P N and let us denote the projection map G ÞÑ H by p. Suppose that tGiuiě1 is a ﬁltration of normal subgroups of G with Property (2). Then, tppGiquiě1 is a ﬁltration of normal subgroups of H with Property (2).

m Proof. Let us denote Z by A and let us deﬁne Hi to be ppGiq. Thus, AGi “ AHi. Let C be a ﬁnitely generated abelian subgroup of H. Note that the group ApHi X Cq is contained in

AHi X AC “ AGi X AC “ ApGi X ACq.

But since the ﬁltration tGiu has Property (2), there is i1 “ ipACq such that Gi X AC “ 1. To sum up, 1 since A X C Ď A X H “ 1, ApHi X Cq being equal to A implies that Hi X C “ 1 for i ě i1.

Proposition 3.6. Let G be a Droms RAAG and tGiuiě1 be a ﬁltration of normal subgroups of G with

Property (2). Then, for suﬃciently large i0 the group Gi0 is free.

Proof. Let G be a Droms RAAG of level lpGq. If lpGq “ 0, then G is a free group, so the statement m clearly holds. If lpGqě 1, then G equals Z ˆ pK1 ˚¨¨¨˚ Kkq for some m P N Y t0u and K1,..., Kk are Droms RAAGs of level less than lpGq. m Let us denote Z and K1 ˚¨¨¨˚ Kk by A and H, respectively, and the projection map G ÞÑ H by p.

By hypothesis, there is i1 “ ipAq such that A X Gi1 “ 1.

Let us deﬁne Hi to be ppGiq. From Lemma 3.5 we get that there is i2 ě 1 such that tHiuiěi2 is a

ﬁltration of H with Property (2), so for i3 “ maxti1, i2u the ﬁltration tHiuiěi3 of normal subgroups of H has Property (2) and Gi » Hi.

The group Hi is a free product of conjugates of Hi X Kj for j P t1,..., ku and a free group. For j P t1,..., ku, tHi XKjuiěi3 is a filtration of normal subgroupsof Kj with Property (2), so by inductive hypothesis, there is ij such that Hij X Kj is free. In conclusion, by taking i0 to be maxti1,..., iku, Hi0 is a free group. 7 Theorem 3.7. Let G be a Droms RAAG with filtration tGiuiě1 of normal subgroups such that G{Gi is torsion-free and iě1 Gi “ 1. Then, for sufficiently large i0 the group Gi0 is free. Ş Proof. It follows from Lemma 3.4 and Proposition 3.6.

In order to show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent), we will show that ICE groups over Droms RAAGs are free-by-(torsion-free nilpotent). A limit group over Droms RAAG is a ﬁnitely generated subgroup of an ICE group over Droms RAAG, so it will also be free-by-(torsion-free nilpotent). Let us start recalling the deﬁnition of ICE groups over Droms RAAGs given in [10].

Deﬁnition 3.8. Let H be a group and Z Ď H the centralizer of an element. Then, the group n G “ H ˚Z pZ ˆ Z q is said to be obtained from H by an extension of a centralizer. An ICE group over H is a group obtained from H by applying ﬁnitely many times the extension of a centralizer construction.

Theorem 3.9 ([10]). All ICE groups over Droms RAAGs are limit groups over Droms RAAGs. Moreover, a group is a limit group over Droms RAAGs if and only if it is a ﬁnitely generated subgroup of an ICE group over Droms RAAGs.

Remark 3.10. If G is a Droms RAAG such that lpGq “ 0, then the centralizer of any non-trivial m element is an inﬁnite cyclic group. Now let G be a Droms RAAGof the form Z ˆpG1 ˚¨¨¨˚Gnq for m, n P N Y t0u such that lpGiqď lpGq´ 1 for i P t1,..., nu, let 1 P G and let us denote the centralizer of 1 in G by CGp1q. m Firstly, if 1 P Z , then CGp1q“ G, so the extension of centralizer construction gives Zk Zm Zk G ˚CGp1q pCGp1qˆ q » p ˆ q ˆ pG1 ˚¨¨¨˚ Gnq.

m Secondly, assume that 1 P Z 10, 10 , 1G and 10 is an elliptic element in G1 ˚¨¨¨˚ Gn. Then, G 10 P 1ďiďn Gi , CGp1q “ CGp10q and we can assume without loss of generality that 1 “ 10. Since extensionsŤ of centralizers are conjugate invariant, we can assume that 1 P Gi for some i P t1,..., nu. Thus, m CGp1q“ Z ˆ CGi p1q. Therefore, k m k G ˚ p q pCGp1qˆ Z q» Z ˆ G1 ˚¨¨¨˚ Gi´1 ˚ Gi ˚ p q pZ ˆ CG p1qq ˚ Gi`1 ˚¨¨¨˚ Gn . CG 1 CGi 1 i ´ ` ˘ ¯ That is, this extension is constructed by extending the centralizer of 1 in Gi and then doing the free product operation and adding the center Zm. m Thirdly, assume that 1 P Z 10, 10 , 1G and 10 is a hyperbolic element in G1 ˚¨¨¨˚ Gn. Then, G 10 P pG1 ˚¨¨¨˚ Gnqzp 1ďiďn Gi q, CGp1q“ CGp10q and we can assume without loss of generality that m m m`1 1 “ 10. Then, CGp1q“Ť Z ˆ x1y» Z ˆx11y» Z and ak m m k Z Z m Z Z G ˚CGp1q pCGp1qˆ q» ˆ pG1 ˚¨¨¨˚ Gnq ˚Z ˆx11y p ˆx11yˆ q» ´ ¯ m k`1 Z ˆ pG1 ˚¨¨¨˚ Gnq ˚Z Z . ´ ¯ 1 1 m 1 m In all the cases we obtain a group of the type Z ˆ pH1 ˚¨¨¨˚ Hnq for m ě m or H “ Z ˆ pA ˚B Cq for m1 “ m. If we continue applying the extension of a centralizer construction for the centralizer 8 1 m of some element h, we can consider as before the cases when h P Z Y H1 ˚¨¨¨˚ Hn Y A ˚B C and in the case when h P H1 ˚ ... ˚ Hn or h P A ˚B C, h can be either elliptic or hyperbolic with respect to that decomposition. Moreover, in [10, Lemma 7.5] it is proved that any ICE over a Droms RAAG can be obtained by ﬁrst extending centralizers of central elements (which gives again a Droms RAAG), then extending centralizers of elliptic elements and ﬁnally considering extensions of centralizers of hyperbolic elements. Thus, the class of ICE over Droms RAAGs can be given the following hierarchical structure.

Deﬁnition 3.11. Assume that G is a Droms RAAG of level 0, that is, G is a free group. An ICE group over G of level 0 is precisely G. An ICE group over G of level k ě 1 is an amalgamated free product over Zn0 of an ICE group over G of level ď k ´ 1 and a free abelian group Zn0 ˆ Zm0 , where Zn0 embeds in Zn0 ˆ Zm0 naturally. m Assume that G is a Droms RAAG of level l ě 1, that is, G “ Z ˆ pG1 ˚¨¨¨˚ Gnq where each Gi is a 1 m Droms RAAG of level less than l. An ICEgroup over G oflevel 0is oftheform Z ˆ pH1 ˚¨¨¨˚ Hnq 1 where m ě m and Hi is an ICE group over Gi for i P t1,..., nu. An ICE group over G of level k ě 1 is an amalgamated free product over Zn0 of an ICE group over G of level ď k ´ 1 and a free abelian group Zn0 ˆ Zm0 , where Zn0 embeds in Zn0 ˆ Zm0 naturally.

Theorem 3.12. Let G be a Droms RAAG and let K be an ICE group over G. Let tKiuiě1 be a ﬁltration of normal subgroups of K with Property (2). Then, for suﬃciently large i0 the group Ki0 is free.

Proof. We prove it by induction on the level lpGq of G as a Droms RAAG. Suppose that lpGq “ 0. Then, if K has level 0, K is precisely G which is a free group, and the result is obvious. If K has level ě 1, then K is of the form m0 n0 H ˚Zn0 pZ ˆ Z q, and H is an ICE group over G of smaller level than K. Since Ki is a normal subgroup of K, Ki inherits a graph of groups decomposition where the vertex groups are of the form

1αj m0 n0 1αj pKi X Hq or pKi X pZ ˆ Z qq , for 1αj P K, and the edge groups are of the form

n0 1βj pKi X Z q for 1βj P K.

Note that tKi XHuiě1 is a filtration of H with Property(2), so by inductive hypothesis,there is i1 such Zm0 Zn0 that Ki1 X H is free. In addition, the filtration tKiuiě1 has Property (2), so there is i2 “ ip ˆ q m0 n0 n0 such that Ki2 X pZ ˆ Z q is trivial. In particular, Ki2 X Z is also trivial. Therefore, taking i0 to be maxti1, i2u, Ki0 is free. Now suppose that lpGqě 1. Then, G is of the form m Z ˆ pG1 ˚¨¨¨˚ Gnq, for some m P N Y t0u and Gi is a Droms RAAG with lpGiq ď lpGq´ 1 for each i P t1,..., nu. If K has level 0 as an ICE group over G, then 1 m K “ Z ˆ pH1 ˚¨¨¨˚ Hnq, 1 where m ě m and Hi is an ICE group over Gi for i P t1,..., nu. Let us denote the projection map K ÞÑ H1 ˚¨¨¨˚ Hn by p. 9 1 1 Zm Zm By hypothesis, there is i1 “ ip q such that X Ki1 “ 1. In addition, by Lemma 3.5, tNiuiě1 is a filtration of normal subgroups of H1 ˚¨¨¨˚ Hn with Property (2), where Ni “ ppKiq. Asa consequence, Ni » Ki for i ě i1.

Note that the group Ni is a free product of conjugates of Ni X Hj for j P t1,..., nu and a free group. For j P t1,..., nu, tNi X Hjuiě1 is a ﬁltration of normal subgroups of Hj with Property (2), so by inductive hypothesis, there is rj such that Nrj X Hj is free. In conclusion, taking i0 to be maxti1, r1,..., rnu, Ni0 is a free group. Finally, suppose that K is an ICE group over G of level k ě 1. Then, K is an amalgamated free product over Zn0 of an ICE group over G of level ď k ´ 1 and a free abelian group Zn0 ˆ Zm0 . This case may be treated as the case when K is an ICE group of level greaterthan 0 overa free group.

Recall that γipGq denotes the lower central series of a group G. That is,

γ1pGq“ G and γi`1pGq “ rG,γipGqs. We denote by torpGq the set of torsion elements of G. In the case when G is ﬁnitely generated nilpotent, then xtorpGqy is the maximal ﬁnite subgroup of G.

Theorem 3.13. Let G be a Droms RAAG, let K be an ICE group over G and deﬁne Ki to be Ki{γipKq “ xtorpK{γipKqqy. Then, Ki`1 ă Ki,Ki is normal in K, K{Ki is torsion-free nilpotent and iě1 Ki “ 1. Ş Proof. By construction, Ki is a characteristic subgroup of K and Ki`1 ă Ki. It remains to show that i Ki “ 1. Suppose that k P p i Kiqzt1u. Since K is a limit group over G, there is a homomorphism Şϕ: K ÞÑ G such that ϕpkq , 1.Ş Let G0 “ impϕq, so G0 is a Droms RAAG. By [23, Theorem 6.4], we have that G0{γipG0q is torsion-free, hence ϕpKiq“ ϕpγipKqq. Then ϕpkq P i ϕpKiq“ i ϕpγipKqq Ď i γipGq“ 1, a contradiction. Ş Ş Ş Corollary 3.14. Every ICE group over a Droms RAAG is free-by-(torsion-free nilpotent).

Proof. It follows from Lemma 3.4, Theorem 3.12 and Theorem 3.13.

Corollary 3.15. Every limit group over Droms RAAGs is free-by-(torsion-free nilpotent).

Proof. It follows from Theorem 3.9 and Corollary 3.14.

4. Corollaries on subdirect products of limit groups over Droms RAAGs In this sectionwe aim to prove TheoremB.TheoremBis a generalisation of [19, Theorem 11].

Theorem 4.1. [19, Theorem 11] Let G1,..., Gm be non-abelian limit groups over free groups and let

S ă G ≔ G1 ˆ¨¨¨ˆ Gm be a finitely generated subdirect product intersecting each factor Gi non-trivially. Suppose further that, for some fixed natural number s P t2,..., mu, for every subgroup S0 of finite index in S the homology group HipS0, Qq is finite dimensional (over Q) for all i ď s. Then for every canonical projection

pj1,...,js : S ÞÑ Gj1 ˆ¨¨¨ˆ Gjs the index of pj1,...,js pSq in Gj1 ˆ¨¨¨ˆ Gjs is ﬁnite. 10 We observe that in [19] the domain of pj1,...,js is deﬁned to be G and not as we stated it above, but in [6] and [7] it is S. It looks more convenient to stick to the latter. The starting point in the proof of Theorem B is that the proof of Theorem 4.1 applies in bigger generality. We explain this in the following result.

Theorem 4.2. Let 2 ď s ď m be integers and S ă G1 ˆ¨¨¨ˆ Gm be a ﬁnitely generated subdirect product such that:

(1) there exist normal free subgroups Li in Gi with Qi ≔ Gi{Li nilpotent;

(2) each Gi is ﬁnitely presented;

(3) for every 1 ď j1 ă¨¨¨ă js ď mifMj1,...,js is a subgroup of inﬁnite index in Gj1 ˆ¨¨¨ˆ Gjs , then there Q Q exists i ď s such that HipMj1,...,js , q is inﬁnite dimensional over ;

(4) for every 1 ď j1 ă¨¨¨ă js ď m and every subgroup Hji of finite index in Gji we have that if a subdirect product Mj1,...,js Ď Hj1 ˆ¨¨¨ˆ Hjs is finitely presented, then there is a subgroup Kji of finite index in Hji and Nji a normal subgroup of Hji such that Kji {Nji is nilpotent and Nj1 ˆ¨¨¨ˆ Njs Ď Mj1,...,js ; (5) S virtually surjects on pairs;

(6) L ≔ L1 ˆ¨¨¨ˆ Lm Ď S;

(7) for every subgroup S0 of ﬁnite index in S, the homology group HipS0, Qq is ﬁnite dimensional (over Q) for all i ď s. Then for every canonical projection

pj1,...,js : S ÞÑ Gj1 ˆ¨¨¨ˆ Gjs , the index of pj1,...,js pSq in Gj1 ˆ¨¨¨ˆ Gjs is ﬁnite.

Proof. We will prove the result by induction on s. For s “ 2, this is condition 5. Suppose that s ě 3. We divide the proof in several steps.

1) Set Q ≔ S{L ă Q1 ˆ¨¨¨ˆ Qm. Consider the Lyndon–Hochschild–Serre spectral sequence 2 Q Ei,j “ HipQ, HjpL, qq that converges to Hi`jpS, Qq.

Note that since each Li is a free group, for every t ď m we have that Q Q Q (1) HtpL, q» H1pLj1 , qbQ ¨¨¨bQ H1pLjt , q, 1ďj1ăj2àă¨¨¨ăjtďm where each summand is Q-invariant, where the Q-action is induced by conjugation. Thus,

2 Q Q Q Q E0,s “ H0pQ, HspL, qq » H1pLj1 , qb ... b H1pLjs , qbQQ . 1ďj1ăj2àă...ăjsďm ´ ¯

By inductive hypothesis, S virtually surjects onto s ´ 1 factors. This implies that HjpL, Qq is a finitely generated ZQ-module for j ď s ´ 1, and hence, 2 Q (2) Ei,j is finite dimensional over for every j ď s ´ 1. 11 i Q For i ě 2, we have that s ` 1 ´ i ď s ´ 1, so by (2), Ei,s`1í is finite dimensional over . Hence, for ff i i i i Q all di erential maps di,j : Ei,s`1í ÞÑ E0,s, impdi,jq is finite dimensional over . Thus i`1 i i i i E0,s “ kerpd0,sq{impdi,jq“ E0,s{impdi,jq Q i Q 2 is finite dimensional over if and only if E0,s is finite dimensional over . This implies that E0,s is Q 8 Q finite dimensional over if and only if E0,s is finite dimensional over . Combining this with the convergence of the spectral sequence and (2) we deduce that 2 HspS, Qq is finite dimensional over Q if and only if E0,s “ H0pQ, HspL, Qqq is finite dimensional over Q.

The condition that HspS, Qq is finite dimensional over Q implies that for each 1 ď j1 ă j2 ă ... ă js ď m, Q Q Q (3) dimQ H1pLj1 , qb¨¨¨b H1pLjs , qbQQ “ ` ˘ dimQ H1pLj , Qqb¨¨¨b H1pLj , QqbQ p q Q ă8, 1 s hj1,...,js Q ` ˘ where hj1,...,js : Q ÞÑ Qj1 ˆ¨¨¨ˆ Qjs is the canonical projection. ≔ 2) We consider S a subgroup of finite index in pj1,...,js pSq that contains L Lj1 ˆ¨¨¨ˆ Ljs and we set ≔ Q S{L. Then,r the Lyndon–Hochschild–Serre spectral sequence r 2 Q r r r Ei,j “ HipQ, HjpL, qq Q r r r 2 2 converges to Hi`jpS, q. By (1) and (3) we deduce that dimQ Ei,j ă8 for j ď s´1 and dimQ E0,s ă8. Then, by the convergencer of the spectral sequence, we deducer that dimQ HipS, Qqă8 forri ď s. r 3) Now we consider S0 an arbitrary subgroup of finite index in pj1,...,js pSq. We view pj1,...,js pSq as a subdirect product of G ˆ¨¨¨ˆ G . As S virtually surjects on pairs we have that p pSq virtually rj1 js j1,...,js surjectsonpairs, soby[6, TheoremA], pj1,...,js pSq is finitely presented. Hence S0 is finitely presented. p qˆ¨¨¨ˆ p q Then, by condition 4 applied to S0 considered as a subdirect product of pj1 Sr0 pjs S0 , there p q ffi p qĎ is a subgroup of finite index Kji rin pji S0 such that for su ciently big t wer have that γt Krji S0. By condition 1, we can choose t sufficiently big so that γ pG qĎ L is free for every 1 ď k ď m. In r t k k r particular, γtpKji q is free.

For every j P t1,..., muztj1,..., jsu we set Kj tobe Gj. Then, S0 ≔ SXpK1 ˆ¨¨¨ˆKmq is a subgroupof ﬁnite index in S such that γtpK1qˆ¨¨¨ˆ γtpKmqĎ S0. By condition 7, HipS0, Qq is ﬁnite dimensional (over Q) for all i ď s. Applying Step 2 for S substituted with S0, Gj substituted with Kj and Lj substituted with γtpKjq, we deduce that Q dimQ Hippj1,...,js pS0q, qă8 for i ď s.

But this combined with condition 3 implies that pj1,...,js pS0q has ﬁnite index in Kj1 ˆ¨¨¨ˆ Kjs .

Since S0 Ď S and each Kji has ﬁnite index in Gji , we deduce that pj1,...,js pSq has ﬁnite index in Gj1 ˆ¨¨¨ˆ Gjs .

Let n ě 0, R be an associative ring with identity element and let M be a (right) R-module. Then M is of (homological) type FPn if there is a projective resolution

P: ¨ ¨ ¨ ÞÑ Pn ÞÑ ¨ ¨ ¨ ÞÑ Pi ÞÑ Pi´1 ÞÑ ¨ ¨ ¨ ÞÑ P0 ÞÑ M ÞÑ 0 such that each projective R-module Pi is finitely generated for 0 ď i ď n. A group G is of type FPn over a commutative ring S if the trivial SG-module S is of type FPn over the group algebra SG. 12 Lemma 4.3. [19, Lemma 6] Let Q1,..., Qi be finitely generated nilpotent groups and Vj be a finitely generated QQj-module such that Vj contains a cyclic non-zero free QQj-submodule Wj for 1 ď j ď i. Suppose that Q is a subgroup of Q ≔ Q1 ˆ¨¨¨ˆ Qi such that V1 bQ ¨¨¨bQ Vi is finitely generated as a Q Q-module. Then,r Q has finite index in Q. Propositionr 4.4. [19,r Proposition 7] Let G be a group of negative Euler characteristic χpGq such that the trivial QG-module Q has a free resolution with finitely generated modules and finite length. Then, for any normal subgroup M of G such that Q ≔ G{M is torsion-free nilpotent and M is free, the QQ-module V “ M{rM, MsbZ Q has a non-zero free QQ-submodule, where Q acts on M “ M{rM, Ms via conjugation.

Theorem 4.5. Let S ă G1 ˆ¨¨¨ˆ Gm be a ﬁnitely generated subdirect product such that:

(1) for each 1 ď i ď m the trivial QGi-module Q has a free resolution of ﬁnite length with ﬁnitely generated modules;

(2) for each 1 ď i ď m there is a normal free subgroup Li of Gi such that Gi{Li is torsion-free nilpotent;

(3) L1 ˆ¨¨¨ˆ Lm Ď S;

(4) S is of type FPs over Q;

(5) χpGiqă 0.

Then, for every canonical projection pj1,...,js : S ÞÑ Gj1 ˆ¨¨¨ˆ Gjs the index of pj1,...,js pSq in Gj1 ˆ¨¨¨ˆ Gjs is ﬁnite.

Proof. Note that since Gi is of type FP8 over Q, it is FP1 over Q and so Gi is finitely generated. By Proposition 4.4, Vi ≔ pLi{rLi, Lisq bZ Q has a non-zero free QQi-submodule, where Qi ≔ Gi{Li acts via conjugation. Let F : ... Ñ Fi Ñ Fi´1 Ñ ... Ñ F0 Ñ Q Ñ 0 be a free resolution of the trivial QS-module Q with Fi finitely generated for i ď s. We define L to be L1 ˆ¨¨¨ˆ Lm and since each Li is free, by the Kunneth formula, Q Q HspL, q“ Lj1 {rLj1 , Lj1 sbZ ¨¨¨bZ Ljs {rLjs , Ljs sbZ » Vj1 bQ ¨¨¨bQ Vjs . 1ďj1ă¨¨¨ăà jsďm 1ďj1ă¨¨¨ăà jsďm Note that HspL, Qq» HspF bQL Qq“ kerpdsq{impds`1q, where dj : Fj bQL Q ÞÑ Fj´1 bQL Q is thedifferential of F bQL Q. Since Fs bQL Q is a finitely generated QQ-module, where Q ≔ S{L is afinitely generated nilpotent group and QQis aNoetherianring,we deduce that kerpd q is a finitely generated QQ-module. In particular, H pL, Qq and V b ¨¨¨b V p s p s p j1 Q Q js Q b ¨¨¨b are finitely generated Q-modules. Note thatp the action of Q on Vj1 Q Q Vjs factors through ≔ p q{ ˆ¨¨¨ˆ ˆ¨¨¨ˆ Q pj1,...,js S Lj1 p Ljs . Then, by Lemma 4.3, Q has finitep index in Qj1 Qjs . This is equivalent to p pSq having finite index in G ˆ¨¨¨ˆ G . r j1,...,js j1 r js

The next step is to prove that limit groups over Droms RAAGs satisfy the conditions of Theorem 4.5.

Lemma 4.6. LetG be aDroms RAAGand Γ a limit group over G. Then, χpΓqď 0. Furthermore, χpΓq“ 0 if and only if Γ has non-trivial center. The latter happens precisely when Γ “ Zl ˆ Λ for some l ě 1. 13 Proof. Let us prove it by induction on the level lpGq of G. If lpGq“ 0, then G is a free group, so the result follows from [19, Lemma 5]. Now assume that lpGqě 1. Then, G equals m Z ˆ pG1 ˚¨¨¨˚ Gnq, where m P N Y t0u and Gi is a Droms RAAG such that lpGiq ď lpGq´ 1, for i P t1,..., nu. From l Proposition 2.4 we get that Γ is of the form Z ˆ Λ where Λ is a limit group over G1 ˚¨¨¨˚ Gn and if m “ 0, then l “ 0. Therefore, χpΓq“ χpZlqχpΛq, so if l ě 1, then χpΓq“ 0. Let us compute χpΛq. If the height of Λ is 0, i.e. hpΛq“ 0, then

Λ “ A1 ˚¨¨¨˚ Aj, where for each t P t1,..., ju, At is a limit group over Gi for some i P t1,..., nu. Hence,

χpΛq“ χpAtq ´ pj ´ 1q, tPtÿ1,...,ju so applying the inductive hypothesis, we get that χpΛq ď 1 ´ j. If j ě 2, then χpΛq ă 0. If j “ 1, χpΛq“ χpA1q and A1 is a limit group over Gi. Thus, by induction, χpA1qď 0 and χpA1q“ 0 if and only if A1 has non-trivial center. If hpΛqě 1, then Λ acts cocompactly on a tree T where the edge stabilizers are cyclic and the vertex groups are limit groups over G1 ˚¨¨¨˚ Gn of height at most hpΛq´ 1. Moreover, at least one vertex group Hv0 has trivial center and so by inductive hypothesis, χpHv0 qă 0. If X is the quotient graph T{Λ,

χpΛq“ χpHvq´ χpHeqď χpHvqď χpHv0 qă 0. vPÿVpXq ePÿEpXq vPÿVpXq

Lemma 4.7. Let Γ be a limit group over Droms RAAGs. Then, the trivial QΓ-module Q has a free resolution with ﬁnitely generated modules and of ﬁnite length.

Proof. Limit groups over Droms RAAGs are of type FP8 over Z (and so over Q) and of ﬁnite cohomological dimension.

Proof of Theorem B.

By [21, Theorem 6.2], Gi{pS X Giq is virtually nilpotent for every i P t1,..., mu. By substituting

Gi and S with subgroups of finite index if necessary we can assume that γmi pGiq Ď S for some ≔ mi. By Proposition A , for some ni ě mi we have that Li γni pGiq is free, Gi{Li is nilpotent and Li Ď S. Now by substituting again Gi and S with subgroups of finite index if necessary, but without changing Li, we can assume that Li is a normal subgroup of Gi such that Li is free, Gi{Li is torsion-free nilpotent and L1 ˆ¨¨¨ˆ Lm Ď S. Note that we cannot assume anymore that Li is γni pGiq but conditions 1 and 6 from Theorem 4.2 hold. The other conditions from Theorem 4.2 hold when each Gi is a limit group over Droms RAAGs with trivial center: condition 2 is [10, Corollary 7.8], condition 3 is Theorem 2.6, condition 5 is Theorem 2.7. Observe that a finite index subgroup of a limit group over Droms RAAGs is a limit group over Droms RAAGs. Hence condition 4 follows from [21, Theorem 6.2]. Finally, condition 7 is assumed in the statement. 14 5. On the L2-Betti numbers and volume gradients of limit groups over Droms RAAGs and their subdirect products

The aim of this section is to study the growth of homology groups and the volume gradients for limit groups over Droms RAAGs and for finitely presented residually Droms RAAGs, following the paper [8]. Some of the results concerning limit groups over Droms RAAGs hold in a more general setting, more precisely, they also hold for limit groups over coherent RAAGs (see Section 5.2 for the definition). Thus, these results (see Theorem C and Theorem D) will be stated for limit groups over coherent RAAGs. However, the results for finitely presented residually Droms RAAGs make use of Theorem 2.7 and in [13] it was shown that this no longer holds for coherent RAAGs. Thus, Theorem E and Theorem F are stated just for residually Droms RAAGs. In order to study the homology growth and volume gradients, we work with exhausting normal chains: a chain pBnq of normal subgroups of finite index such that Bn`1 Ď Bn and n Bn “ 1. Note that if a group is residually finite, then it has an exhausting normal chain. InŞ particular, limit groups over coherent RAAGs have exhausting normal chains.

Given a group G of homotopical type Fm, volmpGq is defined to be the least number of m-cells among all classifying spaces KpG, 1q with finite m-skeleton. For instance, vol1pGq equals dpGq, where dpGq is the minimal number of generators of G. One of the aims of this sectionis to prove Theorem C and Theorem D. These two results are proved in [8] for limit groups over free groups via more technical results that make use of slowness of limit groups over free groups (see Section 5.1 for the definition). In [8] it is shown that limit groups over free groups are slow above dimension 1 and hence are K-slow above dimension 1. Thus, the key point is to show that limit groups over coherent RAAGs are also slow above dimension 1 (see Section 5.2).

Theorem 5.1. [8, Theorem D] If a residually ﬁnite group G of type F is slow above dimension 1, then with respect to every exhausting normal chain pBnq, (1) Rank gradient: dpBnq RGpG, pBnqq “ lim “´χpGq. nÑ8 rG: Bns (2) Deﬁciency gradient: defpBnq DGpG, pBnqq “ lim “ χpGq. nÑ8 rG: Bns

Lemma 5.2. [8, Lemma 5.2] Let K be a ﬁeld and let G be a residually ﬁnite group of type F with an exhausting normal chain pBnq. If G is K-slow above dimension 1, then

dim H1pBn, Kq lim “´χpGq. nÑ8 rG: Bns

We state other results from [8] that will be important for us, as we will show that residually Droms RAAGs G that are of type FPm for some m ě 2 satisfy virtually the assumptions of Theorem 5.3.

Theorem 5.3. [8, Theorem F] Let G Ď G1 ˆ¨¨¨ˆ Gk be a subdirect product of residually finite groups of type F, each of which contains a normal free subgroup Fi ă Gi such that Gi{Fi is torsion-free and nilpotent. Assume Fi Ď G X Gi. Letm ă k be an integer, let K be a field, and suppose that each Gi is K-slow above dimension 1. 15 If the projection of G to each m-tuple of factors Gj1 ˆ¨¨¨ˆ Gjm is of finite index, then there exists an exhausting normal chain pBnq in G so that for 0 ď j ď m,

dim HjpBn, Kq lim “ 0. nÑ8 rG: Bns

Theorem 5.4. [8, Theorem G] Every ﬁnitely presented residually free group G that is not a limit group over free groups admits an exhausting normal chain pBnq with respect to which the rank gradient

dpBnq RGpG, pBnqq “ lim “ 0. nÑ8 rG: Bns

Furthermore, if G is of type FP3 but it is not commesurable with a product of two limit groups over free groups, pBnq can be chosen so that the deﬁciency gradient DGpG, pBnqq “ 0.

5.1. Preliminaries on groups that are K-slow above dimension 1 and slow above dimension 1. Let us start recalling the deﬁnitions from [8] about slowness and K-slowness.

Deﬁnition 5.5. Let G be a group. A sequence of non-negative integers prjqjě0 is a volume vector for G if there is a classifying space KpG, 1q that, for all j P N, has exactly rj j-cells.

Definition 5.6. A group G of homotopical type F is slow above dimension 1 if it is residually finite and for every exhausting normal chain pBnq, there exist volume vectors prjpBnqqj for Bn with finitely many non-zero entries, so that rjpBnq lim “ 0, nÑ8 rG: Bns for all j ě 2. G is slow if it satisfies the additional requirement that the limit exists and is zero for j “ 1 as well.

Example 5.7. [8, Examples 4.4] Finitely generated torsion-free nilpotent groups are slow. The trivial group is slow. Free groups are slow above dimension 1. Surface groups are slow above dimension 1.

Proposition 5.8. [8, Proposition 4.5] If a residually ﬁnite group G is the fundamental group of a ﬁnite graph of groups where all of the edge groups are slow and all of the vertex groups are slow above dimension 1, then G is slow above dimension 1.

Definition 5.9. Let K be a field and let G be a residually finite group. G is K-slow above dimension 1 if for every exhausting normal chain pBnq, we have

dimK HjpBn, Kq lim “ 0, nÑ8 rG: Bns for all j ě 2. G is K-slow if it satisﬁes the additional requirement that the limit exists and is zero for j “ 1 as well.

It follows directly by the deﬁnitions that if a group G is slow above dimension 1 (respectively, slow), then it is K-slow above dimension 1 (respectively, K-slow).

Proposition 5.10. [8, Proposition 5.3] Let K be a field. If a residually finite group G is the fundamental group of a finite graph of groups where all of the edge groups are K-slow and all of the vertex groups are K-slow above dimension 1, then G is K-slow above dimension 1. 16 5.2. Limit groups over coherent RAAGs are slow above dimension 1. Recall that a group is coherent if all its finitely generated subgroups are also finitely presented. In particular, Droms RAAGs are coherent RAAGs.

Lemma 5.11. Coherent RAAGs are slow above dimension 1. In particular, Droms RAAGs are slow above dimension 1.

Proof. In [16] Droms proved that if GX is a coherent RAAG, then GX splits as a ﬁnite graph of groups where all the vertex groups are free abelian. Thus, by Proposition 5.8, coherents RAAGs are slow above dimension 1.

Thegoalofthissectionistoshowthatlimit groupsovercoherent RAAGs are slow above dimension 1. For that, we will use the work on [10] and [11] about limit groups over coherent RAAGs. Let us recall some deﬁnitions that are needed to understand the results below. If GX is a RAAG, the elements of X are called the canonical generators of GX. A non-exceptional surface is a surface which is not a non-orientable surface of genus 1, 2 or 3.

Proposition 5.12. [10, Lemma 7.3] Let G be a coherent right-angled Artin group. A graph tower over G of height 0 is a coherent RAAG H which is obtained from G by extending centralisers of canonical generators of G. A graph tower over G of height ě 1 can be obtained as a free product with amalgamation, where the edge group is a free abelian group, one of the vertex groups is a graph tower over G of lower height and the other vertex group is either free abelian, or the direct product of a free abelian group and the fundamental group of a non-exceptional surface.

Theorem 5.13. [11, Theorem 8.1] Let Γ be a limit group over a RAAG G. Then, Γ is a subgroup of a graph tower over G.

We now prove some results that will be used in order to show that limit groups over coherent RAAGs are slow above dimension 1.

Lemma 5.14. [24] Let 1 Ñ C Ñ D Ñ E Ñ 1 be a short exact sequence of groups of type F. Suppose that there are classifying spaces KpC, 1q and KpE, 1q with αtpCq and αtpEq t-cells, respectively. Then, there is a KpD, 1q complex with αipDq i-cells such that

αipDq“ αtpCqαi´tpEq. 0ďtďi ÿ Lemma 5.15. Suppose that G is a group of homotopical type F, H is a normal subgroup of ﬁnite index in G and prjpGqqj is a volume vector for G. Then, prG: HsrjpGqqj is a volume vector for H.

Proof. Let Y be a classifying space KpG, 1q that, for all j P N Y t0u, has exactly rjpGq j-cells. Then, if we denote by Y the universal cover of Y, Y is contractible and Y “ Y{G. Therefore, Y{H is a classifying space for H with exactly rG: Hsr pGq open j-cells. r j r r r Lemma 5.16. Let G be a group of homotopical type F and H a group where there is a short exact sequence p 1 Zn HG 1 . a) If n ě 1, then H is slow. 17 b) If n ě 0 and G is slow above dimension 1, then H is slow above dimension 1.

Proof. Let us denote Zn by A. a) Let pBiq be a chain of ﬁnite index normal subgroups in H with Bi “ 1. We need to show that for each i there is a KpBi, 1q complex with rjpBiq j-cells such that forŞ each j ě 0,

rjpBiq lim “ 0. iÑ8 rA ˆ G: Bis For each i, the short exact sequence from the statement induces a short exact sequence

1 A X Bi Bi ppBiq 1. Let us show that rH : Bis “ rA: A X BisrG: ppBiqs. Indeed, note that rH : Bis “ rH : ABisrABi : Bis. Firstly, rABi : Bis equals rA: A X Bis. Secondly, rH : ABis equals rH{A: ABi{As, and H{A » G and ABi{A » ppBiq. Therefore, rH : ABis “ rG: ppBiqs.

Let αjpGq be the number of j-cells in a ﬁxed KpG, 1q complex. By Lemma 5.15, there is a KpppBiq, 1q complex with αjpppBiqq j-cells such that

αjpppBiqq “ rG: ppBiqsαjpGq. n Since A X Bi has ﬁnite index in A, there is a KpA X Bi, 1q complex with j j-cells. By Lemma 5.14, there is a KpBi, 1q complex with αjpBiq j-cells such that ` ˘ n n αjpBiq“ αapppBiqq “ rG: ppBiqsαapGq, 0ďaďj 0ďaďj ÿ ˆj ´ a˙ ÿ ˆj ´ a˙ and we set rjpBiq“ αjpBiq. Then, n rG: ppB qs αapGq rjpBiq i 0ďaďj j´a n 1 lim “ lim ÿ ` ˘ “ αapGq lim “ 0. 0ďaďj iÑ8 rH : Bis iÑ8 rG: ppBiqsrA: Bi X As ÿ ˆj ´ a˙ iÑ8 rA: Bi X As b) If n ě 1, then we apply a). If n “ 0, by assumption G is slow above dimension 1.

Theorem 5.17. Limit groups over coherent RAAGs are slow above dimension 1.

Proof. Let G be a coherent RAAG and let Γ be a limit group over G. Then, by Theorem 5.13, Γ is a subgroup of a graph tower over G, say L. Let us prove by induction on the height of L that Γ is slow above dimension 1 (see Proposition 5.12). If L has height 0, L is a coherent RAAG. Therefore, L is the fundamental group of a graph of groups where the vertex groups are free abelian. Thus, Γ also admits a decomposition as a graph of groups where the vertex groups are free abelian, so by Proposition 5.8 we get that Γ is slow above dimension 1. Now suppose that the height of L is greater than 0. Then, L is a free product with amalgamation, where the edge group is a free abelian group, one of the vertex groups is a graph tower over G of lower height and the other vertex group is either free abelian, or the direct product of a free abelian group and the fundamental group of a non-exceptional surface. Then, Γ admits a decomposition as a graph of groups where the edge groups are free abelian (including the possibility that some are trivial), and the vertex groups are either subgroups of graph towers over G of lower height (and by 18 induction, those vertex groups are slow above dimension 1) or free abelian groups or subgroups of the direct product of a free abelian group and the fundamental group of a non-exceptional surface. It suffices to show that finitely generated subgroups of the direct product of a free abelian group and the fundamental group of a non-exceptional surface are slow above dimension 1. Then, by Proposition 5.8 we obtain that Γ is slow above dimension 1. m If H is a finitely generated subgroup of Z ˆ G0 where m P N Y t0u and G0 is the fundamental group of a non-exceptional surface, then there is a short exact sequence

1 A H N 1, where A is a free abelian group and N is a ﬁnitely generated subgroup of G0. In particular, N is either a surface group or a free group, so N is slow above dimension 1. Then, by Lemma 5.16, H is slow above dimension 1.

Theorem 5.17 implies that limit groups over Droms RAAGs are slow above dimension 1. We observe that if we want to study which RAAGs are K-slow above dimension 1 we can apply [3, Theorem 1] that states that for a right-angled Artin group G with underlying ﬂag complex L and a ﬁeld K, for any exhausting normal chain pBnq in G,

dimK HjpBn, Kq lim “ b¯ j´1pL, Kq, nÑ8 rG: Bns where b¯ j´1pL, Kq denotes the reduced Betti number of L with coeﬃcients in K.

Proof of Theorem C Parts a) and b) follow from Theorem 5.1 and Theorem 5.17. Part c) follows from Theorem 5.17.

Proof of Theorem D It follows from Lemma 5.2 and the fact that Theorem 5.17 implies that every limit group over a coherent RAAG is K-slow above dimension 1.

Proof of Theorem E The proof is an adaptation of the proof from [8] to the setting of residually Droms RAAGs. Thus, we just give a sketch of the proof. From Theorem 2.7 we get that G is a full subdirect product of limit groups over Droms RAAGs G0 ˆ G1 ˆ¨¨¨ˆ Gr such that G0 is abelian (possibly trivial), G X G0 has ﬁnite index in G0 and Gi has trivial center for i P t1, 2,..., ru. By Proposition A, Gi is free-by-(torsion-free nilpotent).

Suppose that G0 is trivial. Then, part (1) from Theorem E follows from Theorem 5.3 and Theorem B. A word of warning is needed here. Though each Gi has a free normal subgroup Ni such that Gi{Ni is torsion-free nilpotent we cannot guarantee that Ni is a subgroup of G. Butby [21, Theorem 6.2], Gi{pG X Giq is virtually nilpotent, hence there is a subgroup of ﬁnite index Gi in Gi and a free normal subgroup Fi of Gi such that Gi{Fi is torsion-free nilpotent and Fi Ď G. Thenr we can apply ≔ Theorem 5.3 for G G X pG1 ˆ¨¨¨ˆr Grqă G1 ˆ¨¨¨ˆ Gr. This guarantees an exhausting normal chain pBnq of G thatr in generalr might notr be ar normal chainr in G.

If G0 is non-trivial,r G X G0 is non-trivial, free abelian and central, so part (1) from Theorem E follows from [8, Lemma 7.2]. It remains to consider part (2) from Theorem E. From Theorem 2.5, G has a subgroup of finite index H “ H1 ˆ¨¨¨ˆ Hr where each Hi is a limit group over Droms RAAGs. Following the 19 proof of [8, Theorem E], we can choose pBiq an exhausting normal chain in H and such that Bi “ pBiH1qˆ¨¨¨ˆpBi X Hrq. Then, as in [8], we can deduce that for j ě 1, p q dim Hj Bn, K r lim “ p´1q δj,rχpGq, nÑ8 rG: Bns where δj,r is the Kronecker symbol. A limit group over Droms RAAGs does not contain a direct product of two or more non-abelian free groups, since limit groups over Droms RAAGs are coherent (see [10, Corollary 7.8]) and the direct product of two non-abelian free groups is not coherent (there are coabelian finitely generated subgroups that are not finitely presented). In addition, every limit group over Droms RAAGs that has trivial center contains a non-abelian free group (see, for instance, [21, Property 2.10]), so r “ ρ unless one or more Hi has non-trivial center. If some Hi has non-trivial center, χpHiq “ 0 and so χpHq“ 0 and χpGq“ 0.

Proof of Theorem F In Theorem 5.4 M. Bridson and D. Kochloukova proved a low dimensional homotopical version of Theorem E for limit groups over free groups. The proof of Theorem 5.4 relies on some general results about residually ﬁnite groups, see [8, Lemma 8.1, Lemma 8.2], and some speciﬁc results for limit groups over free groups. The properties of limit groups over free groups that are used in the proof of Theorem 5.4 still hold for the class of limit groups over Droms RAAGs and correspond to Theorem B and Theorem 2.7, so the same proof can be used in order to show Theorem F.

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