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A fundamental object of study in asymptotic group theory is the Dehn function of a finitely presented group. This relates to other questions in Group theory, such as solvability of the word problem. In this work we address the following question.

Question 1. Which functions can be realised as Dehn functions of K¨ahler groups?

A straightforward search combining known examples of K¨ahler groups with classical results on Dehn functions shows that they include groups with linear, quadratic, and ex- ponential Dehn function (see Section 3). Since all three functions occur as Dehn functions of many lattices in semi-simple Lie groups [34, 26, 35], this is may not be too surprising, considering that many of the known examples of K¨ahler groups arise from such lattices (see e.g. [49]). However, these three functions only cover a small fraction of the functions that can be obtained as Dehn functions of finitely presented groups and they play a very special role among them: hyperbolic groups can be characterised by the property that their Dehn function is linear [30] and there is no non-hyperbolic group with subquadratic Dehn function [30, 7, 43, 20]. In contrast, the set of α ≥ 2 such that there is a group with α Dehn function n is dense in [2, ∞) [8]. Hence, it is natural to ask if K¨ahler groups can attain any Dehn functions other than linear, quadratic and exponential, and the main part of our work will be dedicated to proving that the answer to this question is positive. This is achieved by showing that an example of a K¨ahler subgroup of a direct product of three surface groups, which was constructed by Dimca, Papadima and Suciu [23], has Dehn function bounded below by a cubic function and above by n6.

Theorem 1.1. There is a K¨ahler group G with the following properties: 3 6 (1) G has Dehn function δG with n ≼ δG(n)≼ n ; (2) G is not coherent; (3) G is of type F2 and not of type F3.

It turns out that the group G of Theorem 1.1 can be realised as fundamental group of a smooth projective variety X with Stein universal cover X̃ [23]. To obtain Theorem 1.1 we generalise work by Dison on the Dehn function of subgroups of direct products of free groups [25]. While the upper bound is a straightforward consequence of his work, the lower bound requires some new ideas. Our key estimate to obtain the lower bound is Proposition 4.2, which gives a lower bound on the distorsion of certain elements in a fibre product in terms of the Dehn function of the common quotient. As pointed out to us by the referee, this can be interpreted as a quantitative version of a result of Mihailova [40] proving that there is a subgroup of a direct product of two free groups with unsolvable generalized word problem. We refer the reader to [41, Section 4] for Mihailova’s result and several related unsolvability results for subgroups of direct products of two free groups which can be proved using similar methods. In this context we also want to point out that the finite generation and finite presentability of fibre products in terms of the finiteness properties of the groups involved in their construction has been studied in detail. In particular, there are explicit algorithms → that, given some explicit data and conditions on two homomorphisms φi ∶ Gi Q (i = 1, 2), provide a finite presentation for their fibre product (see [11] and also [3, 24]). 3

Recall that a closed Riemannian manifold with non-positive sectional curvature admits a quadratic isoperimetric function. A natural notion of curvature in the context of K¨ahler manifolds is that of holomorphic bisectional curvature. In contrast with the Riemannian setting, we deduce from Theorem 1.1 the following result. Corollary 1.2. There is a compact K¨ahler manifold with non-positive holomorphic bi- sectional curvature which does not admit a quadratic isoperimetric function. Structure. In Section 2 we summarize results on Dehn functions that we shall need. In Section 3 we provide examples of K¨ahler groups with linear, quadratic and exponential Dehn function. In Section 4 we give a lower bound on the distortion of certain elements in fibre products of hyperbolic groups. We apply this bound in Section 5 to show that there is a K¨ahler group with Dehn function bounded below by a cubic function. In Section 6 we give an upper bound on the Dehn function of this group allowing us to prove Theorem 1.1. We finish by listing some open questions arising from our work in Section 7. Acknowledgments. We would like to thank Pierre Pansu for helpful comments and suggestions and the referee for their valuable remarks.

2. Dehn functions

Let G be a finitely presented group and let G ≅ ⟨X S R⟩ be a finite presentation for G. A word w(X) of length l(w(X)) = m in the alphabet X is an expression of the form ±1 ±1 w(X) = x1 ...xm with xi ∈ X for 1 ≤ i ≤ m. We call a word w(X) null-homotopic in G if it represents the trivial element in G. Every null-homotopic word is freely equal to k 1 1 a word of the form uj X r± uj X − for some words ui X and elements ri R. ∏j=1 ( ) j ( ( )) ( ) ∈ The area of a null-homotopic word w(X) is k ⎧ F X ⎫ ⎪ ( ) ±1 −1⎪ Area(w(X)) = min ⎨k S w(X) = M uj(X)rj (uj(X)) ⎬ ⎪ j 1 ⎪ ⎩⎪ = ⎭⎪ N → R R → R For non-decreasing functions f, g ∶ ≥0 (or ∶ ≥0 ≥0) we say that f is asymptot- ically bounded by g if there are constants C1,C2,C3 ≥ 1 such that f(n) ≤ C1g(C2n)+ C3 for all n ∈ N (or n ∈ R). We write f ≼ g if f is asymptotically bounded by g. We further say that f is asymptotically equal (or asymptotically equivalent) to g and write f ≍ g if f ≼ g ≼ f. For an element g G we denote by g dist 1, g its distance from the origin ∈ S SG = Cay(G,X)( ) in the Cayley graph Cay(G, X) with respect to the generating set X of G. The Dehn function of G is the function

δG(n) = max {Area(w(X)) Sw(X) null-homotopic with l(w(X)) ≤ n} . N → R We say that a function f ∶ >0 is an isoperimetric function for G if δG(n) ≼ f(n). For a subgroup N ≤ G of a finitely generated group G and generating sets X of G and G Y of N, the distortion of N in G is the function ∆N (n) = max {SgSH S g ∈ N, SgSG ≤ n}. G A priori the definitions of S⋅SG and δG, and∆N depend on a choice of a finite presentation for G and a finite generating set for N ≤ G. However, up to asymptotical equivalence, they are independent of these choices and hence it makes sense to speak of them as properties of a finitely presented group rather than of a finite presentation. We want to summarize a few important properties of Dehn functions which we will require. 4

Theorem 2.1. Let G be a finitely presented group and δG be its Dehn function. Then the following hold:

(1) δG is linear if and only if G is hyperbolic; 2 (2) if δG is subquadratic (i.e. δG(n) ≺ n ) then it is linear; (3) if G = π1X for X a closed non-positively curved Riemann manifold or, more generally, if G is CAT(0), then the Dehn function of G is at most quadratic. In 2 particular, if G is not hyperbolic then δG(n) ≍ n ; (4) if G = G1 × G2 is a direct product of two infinite groups then 2 δG(n) ≍ max ™n , δG1 (n), δG2 (n)ž .

Proof. For (1) see [30]. For (2) see [30] and also [43, 7, 20]. For (3) see [10, III.Γ.1.6]. For (4) see [9]. 

3. Linear, quadratic and exponential Dehn functions Comparing the examples of K¨ahler groups in the literature to results on Dehn functions of hyperbolic groups, non-positively curved groups and lattices, it is not hard to see that they include examples with linear, quadratic and exponential Dehn function. In this section we want to give an overview of known examples of K¨ahler groups for which we could determine their Dehn function. Many of these examples rely on a result by Toledo.

Proposition 3.1 ([49]). Let G be a semi-simple Lie group with associated symmetric space X. Assume that X is an irreducible Hermitian symmetric space and that X is neither the one- or two-dimensional complex unit ball, nor the Siegel upper half plane of genus 2. Then every non-uniform lattice Γ ≤ G is a K¨ahler group.

Note that Toledo also shows that non-uniform lattices in SU(1, 1) and SU(2, 1) are not K¨ahler, while it is not known if non-uniform lattices in Sp(4, R) are K¨ahler [49]. Linear Dehn function: By Theorem 2.1(1), classifying K¨ahler groups with linear Dehn function is equivalent to classifying hyperbolic K¨ahler groups. Hyperbolic K¨ahler groups include the fundamental groups Γg = π1Sg of closed orientable surfaces Sg of genus ≥ 2 and cocompact lattices Γ ≤ PU(n, 1), which correspond to compact complex ball quotients n B ~Γ (see [44, Section 2] and also [17, 48] for examples of such lattices). As an immediate consequence we obtain that there are K¨ahler groups with linear Dehn function.

Quadratic Dehn function: It is easy to obtain K¨ahler groups with quadratic Dehn function by taking direct products of hyperbolic groups and applying Theorem 2.1(4). In 2n particular, we obtain that Z and Γg1 ×⋅⋅⋅× Γgr have quadratic Dehn function for n ≥ 1, r ≥ 2, and gi ≥ 2. Searching a bit further we can also find examples of K¨ahler groups with quadratic Dehn function which do not (virtually) decompose as a direct product. They include: ● irreducible non-hyperbolic cocompact lattices in semi-simple Lie groups whose associated symmetric space is Hermitian, since non-compact symmetric spaces are CAT(0) and thus all cocompact lattices have Dehn function bounded above by a quadratic function by Theorem 2.1(3); 5

● the symplectic groups Sp(2g, Z) which have quadratic Dehn function for g ≥ 5 by a result of Cohen [19], and are known to be K¨ahler for g ≥ 3 by Proposition 3.1; H k k ● Heisenberg groups 2k+1 for ≥ 4, which are K¨ahler if and only if ≥ 4 ([14, 47] and [16]) and have quadratic Dehn function for k ≥ 2 [1]; ● the compactifications of non-arithmetic lattices constructed in Py’s work [45], since they are non-positively curved and non-hyperbolic (the fundamental group of the cusps maps to Z2-subgroups in the compactification). A further interesting class of K¨ahler groups with quadratic Dehn function are Hilbert modular groups defined by totally real number fields K~Q of degree n = [K ∶ Q] ≥ 3. More → n precisely, let φ ∶ K R be the homomorphism defined by the n non-trivial embeddings → n of K in R. Then φ defines an embedding PSL(2, K) (PSL(2, R)) and the image of n a cocompact lattice in PSL(2, K) defines a non-uniform lattice in (PSL(2, R)) . Such a lattice is called a Hilbert modular group. It is K¨ahler by [49]. Hilbert modular groups contain Z2 subgroups and are thus not hyperbolic. In particular, their Dehn function can not be linear. On the other hand they are Q-rank one lattices and thus by work of Drutu 2 ǫ [26] have Dehn function bounded above by n + for every ǫ > 0. In fact Drutu’s work shows that the Dehn function of Hilbert modular groups is quadratic (see also [52]). Note that very recently Leuzinger and Young showed that all irreducible non-uniform lattices in a connected center-free semisimple Lie group of R-rank ≥ 3 without compact factors have quadratic Dehn function [35], confirming a Conjecture of Gromov for this very general class of lattices. Hence, any K¨ahler groups of this form also have quadratic Dehn function.

Exponential Dehn function: By work of Leuzinger and Pittet [34], all irreducible non- uniform lattices in semi-simple Lie groups of R-rank 2 have exponential Dehn function. Hence, to see that there are K¨ahler groups with exponential Dehn function it suffices to find an example of a K¨ahler group which can be realised by such a lattice. A class of such examples is given by non-uniform lattices in SU(2,n) for n ≥ 2, since this is a semi-simple irreducible Lie group of real rank 2 (e.g. [32]) and it follows from Proposition 3.1 that non-uniform lattices in SU(2,n) are K¨ahler. The existence of non-uniform lattices in SU(2,n) is well-known. An example is the group SU(2, n, Z[i]), which is a non-uniform lattice by Godement’s compactness criterion [6, 42], since it contains the unipotent element

00 0 00 ⋯ 0 ⎛ 1 1 ⎞ ⎜ 0 1 ⎟ ⎜ 2 2 − ⋮ ⋮ ⎟ ⎜ 1 1 ⎟ ⎜ 0 − 2 − 2 1 ⎟ ⎜ ⎟ In 2 ⎜ 0 1 1 0 0 ⎟ , + + ⎜ − − ⎟ ⎜ ⎟ ⎜ 0 0 0 0 ⎟ ⎜ ⋯ ⋯ ⎟ ⎜ ⎟ ⎜ ⋮ ⋮ ⎟ ⎝ 0 ⋯ 0 ⎠ C where In+2 is the identity matrix in GL(n + 2, ) (for further details see [39, Section 3]). Other Dehn functions: This leaves us with the question if there are any K¨ahler groups whose Dehn function does not fall into one of these three categories. While it is possible that such groups can be found among lattices in semi-simple Lie groups whose associated 6 symmetric space is Hermitian, we are not aware of any known K¨ahler example. Indeed, most of the lattices for which the Dehn function is known seem to fall in one of the previous three categories, which might not be too surprising in the light of the results of Leuzinger and Pittet [34], and Leuzinger and Young [35]. One notable exception is the 3-Heisenberg group which has cubic Dehn function [15, Chapter 8], [28]. However, the latter is not K¨ahler. The remaining sections will be dedicated to proving that K¨ahler groups can admit a Dehn function which is not linear, quadratic or exponential, by proving upper and lower bounds on the Dehn function of a concrete example of a K¨ahler subdirect product of three surface groups constructed by Dimca, Papadima and Suciu [23].

4. Distortion of fibre products of hyperbolic groups For short exact sequences of groups

φ1 1 → N1 → G1 → Q → 1,

φ2 1 → N2 → G2 → Q → 1, their (asymmetric) fibre product P ≤ G1 ×G2 is the group P = {(g1, g2) S φ1(g1) = φ2(g2)}. Recall the well-known 0-1-2-Lemma.

Lemma 4.1 (0-1-2 Lemma). Consider the short exact sequences defined above. If Q is finitely presented and G1 and G2 are finitely generated, then the fibre product P ≤ G1 ×G2 is finitely generated.

The proof is straight-forward by constructing an explicit finite generating set. We will explain its construction in the special case of a symmetric fibre product, that is, G1 = G2 and φ1 = φ2, since we will require it later. The general case is very similar. Let X be a finite generating set for G. Denote by Xi the corresponding generating set ↪ of Gi and by X∆ the one of the diagonal embedding Diag(G) G × G = G1 × G2. Since G is a quotient of the free group FX , the same is true for Q. For notational purposes we denote by XQ the corresponding finite generating set of Q. Since Q is finitely presented we can find a finite set of relations RQ among the elements of XQ, such that Q ≅ ⟨XQ SRQ⟩. Denote by R1 the lift of RQ to words in X1 with respect to the canonical identification X1 ≅ XQ provided by viewing G1 and Q as consecutive quotients of FX = FXQ = FX1 . Since N1 = ker φ1, it follows that N1 = ⟨⟨R1⟩⟩ is finitely generated as normal subgroup of G1. It is now easy to see that a finite generating set of P is given by X∆ ∪ R1. This generating set allows us to give a lower bound for the distortion of P in G × G in terms of the Dehn function δQ of Q with respect to the presentation Q ≅ ⟨XQ SRQ⟩.

Proposition 4.2. Let G = ⟨X S S⟩ be a finitely presented hyperbolic group, let Q be a → finitely presented group and let φ ∶ G Q be an epimorphism. Let Q = ⟨XQ S RQ⟩ be a finite presentation for Q with S⊂R. Let P ≤ G × G be the symmetric fibre product of φ and let hn = (g1,n, 1) ∈ P be a sequence of elements of the intersection P ∩(G1 × {1}) with Sg1,nSG ≍ n. Assume that each g admits a representative word v X with v n 1,n n( ) S n(X )SFree(X ) ≍ such that the null-homotopic word vn(XQ) in Q satisfies Area(vn(XQ)) ≍ δQ(n). Then ShnSP ≽ δQ(n). 7

Proof. If Q is hyperbolic then the conclusion is trivially true, since we have the asymptotic inequalities ShnSP ≥ Sg1,nSG ≍ n ≽ δQ(n), so assume that Q is not hyperbolic. Let G ≅ ⟨X S S⟩ be a finite presentation for G. Choose a finite presentation Q = ⟨XQ S RQ⟩ as above. We denote by S1 ⊂R1 the subset corresponding to the subset S⊂R. Since hn ∈ P ∩ (G1 × {1}), the image φ(hn) = φ1(g1,n) ∈ Q represents the trivial word. There is a word ωn(X∆, R1) with hn = ωn(X∆, R1) in P . Since [G1 × {1} , {1} × G2] = {1}, we obtain ωn(X∆, R1) = ωn(X1, R1) ⋅ ωn(X2, 1). Hence, the word ωn(X2, 1) represents the trivial element in G2. It follows that the word ωn(X1, 1), obtained from ωn(X1, R1) by deleting all occurrences of elements of R1, represents the trivial word in G1. Hence, ωn(X1, 1) is freely equal to a product of finitely many conjugates of elements of S1, whose number will be denoted by k2,n ≥ 0. Since hyperbolic groups have linear Dehn function, we have that for a minimal choice of k2,n, l ω , 1 δ ω , 1 k (4.1) ( n(X1 )) ≽ G1 (S n(X1 )SFree(X1)) ≥ 2,n and thus

l(ωn(X1, R1)) = k1,n + l(ωn(X1, 1)) ≽ k1,n + k2,n, where k1,n ≥ 0 denotes the number of occurences of elements of R1 in ωn(X1, R1). However, it follows from the previous paragraph that ωn(X1, R1) is freely equal to a product of k1,n + k2,n conjugates of elements of R1. Hence, the area of ωn(X1, R1) in Q ≅ ⟨X1 S R1⟩ provides us with a lower bound on k1,n + k2,n and (4.1) yields

l(ωn(X1, R1)) ≽ k1,n + k2,n ≥ AreaQ(ωn(X1, R1)). (4.2) 1 The word ωn(X1, R1) ⋅ (vn(X1))− is null-homotopic in G1. Thus, it can be written as product of k3,n ≥ 0 conjugates of elements of S1, where we choose k3,n to be minimal with this property. Since hyperbolic groups have linear Dehn function, we deduce

k ω , v −1 ω , n. (4.3) 3,n ≼ S n(X1 R1) ⋅ ( n(X1)) SFree(X1) ≼ S n(X1 R1)SFree(X1) +

Using that δQ(n) ≍ AreaQ(vn(X)), we further obtain

AreaQ(ωn(X1, R1)) + k3,n ≥ AreaQ(vn(X1)) ≍ δQ(n) (4.4) Combining inequalities (4.2), (4.3) and (4.4), this yields

δQ(n) ≼ AreaQ(ωn(X1, R1)) + k3,n (4.4)

≼ AreaQ(ωn(X1, R1)) + n + Sωn(X1, R1)SFree X1 (4.3) ( ) ≼ 2 ⋅ l(ωn(X1, R1)) + n. (4.2)

δQ(n) → → Since Q is non-hyperbolic, we obtain that n ∞ as n ∞, and thus l(ωn(X1, R1)) ≽  δQ(n). It follows that ShnSP ≽ δQ(n). A special type of fibre products that we are interested in are the coabelian subgroups of a direct product of groups, that is, subgroups H ≤ G1 ×⋅⋅⋅× Gr with H = ker θ for some → N epimorphism θ ∶ G1 ×⋅⋅⋅× Gr Z . 8

→ N → For a group G consider an epimorphism φ ∶ G Q = Z and denote by φ1 ∶ G1 Q, → φ2 ∶ G2 Q two copies of this epimorphism. Then the coabelian subgroup K = ker (φ1 + φ2) ≤ G1 × G2 is the fibre product of the short exact sequences φ 1 → ker φ → G → Q → 1,

φ 1 → ker φ → G −→ Q → 1. We obtain the following consequence of Proposition 4.2

N Corollary 4.3. Let G be a finitely presented hyperbolic group, let Q = Z for some N ≥ 0 → and let φ ∶ G Q be an epimorphism. Let K be as defined in the previous paragraph and → assume that there is an automorphism ν ∶ G G such that (φ ○ ν)(g) = −φ(g) for all g ∈ G. Let hn = (g1,n, 1) ∈ K be a sequence of element of the intersection K ∩ (G1 × {1}), such that Sg1,nSG = n and g1,n has the same properties as the element g1,n in Proposition 4.2. Then ShnSK ≽ δQ(n). Proof. It is immediate from the existence of the automorphism ν that K is isomorphic → to the symmetric fibre product P of the short exact sequence defined by φ ∶ G Q. → The automorphism is induced by the automorphism (idG,ν) ∶ G1 × G2 G1 × G2 of the product. The element hn is invariant under this automorphism. Hence, Proposition 4.2  implies that ShnSK ≍ ShnSP ≽ δQ(n).

5. Subgroups of direct products of surface groups

1 1 Throughout the remainder of the paper we will fix the convention that [a, b] = aba− b− . Now consider a surface group Γ2 = π1S2 of genus 2 and fix a presentation

Γ2 = ⟨a1, b1, a2, b2 S [a1, b1] [a2, b2]⟩ . Define an epimorphism → 2 φ ∶ Γ2 Z = ⟨a, b S [a, b]⟩ ai ↦ a bi ↦ b

(i) → 2 (i) and take r ≥ 2 copies φi ∶ Γ2 Z , 1 ≤ i ≤ r, where we denote the generators of Γ2 by (i) (i) aj , bj , j = 1, 2. As in Section 4, we define coabelian subgroups Kr = ker θr for r (1) (r) → 2 θr = Q φi ∶ Γ2 ×⋅⋅⋅× Γ2 Z . i=1 It is not hard to see that the groups Kr for r ≥ 3 are in fact explicit realizations of the K¨ahler subgroups of direct products of r surface groups constructed by Dimca, Papadima and Suciu [23] by taking a 2-fold branched cover of genus 2 of an elliptic curve (see [36] for more details and an explicit construction of finite presentations for the Kr). The rest of this section will be concerned with proving the following result:

3 Theorem 5.1. The Dehn function of the K¨ahler group K3 satisfies δK3 (n) ≽ n . 9

It is straight-forward to check that → ν ∶ Γ2 Γ2 ↦ −1 −1 ai (aibi)ai (aibi) (5.1) ↦ −1 −1 bi (aibi)bi (aibi) defines an automorphism of Γ2 which satisfies φ ○ ν = −φ.

2 (1) m (1) m Lemma 5.2. Let Q = Z = ⟨a, b S [a, b]⟩ and let hm = Š(a1 ) , (b2 )  , 1 ∈ K2, for 2 m ≥ 1. Then ShmSK2 ≽ δQ(m)(≍ m ).

(1) m (1) m (1) Proof. Since φ1 Š(a1 ) , (b2 )  = 1, we have hm ∈ K2 ∩ ŠΓ2 × {1}. (1) (1) (1) (1) The group Γ2 retracts onto the free subgroup F2 = ba1 , b2 g ≤ Γ2 via the homo- morphism defined by (1) → Γ2 F2 (1) ↦ (1) a1 a1 (1) ↦ (1) b2 b2 (1) (1) ↦ a2 , b1 1.

(1) Hence, F2 Γ is an undistorted subgroup and in particular we have gm (1) gm F2 ≤ 2 S SΓ2 ≍ S S = (1) m (1) m 4m, where the last inequality is realised by the sequence of words vm(X ) = (a1 ) , (b2 )  with v 4m, for a(1), b(1), a(1), b(1) . It is well-known that δ n S m(X )SFree(X ) = X = š 1 1 2 2 Ÿ Q( ) ≍ 2 m m n and that the sequence of words wm({a, b}) = [a , b ] satisfies AreaQ(wm({a, b})) ≍ δQ(m). Replace the presentation for Q with the presentation

Q Q Q Q Q Q Q Q −1 Q Q −1 Q Q Q Q Q ≅ ba1 , b1 , a2 , b2 S a1 , b1  , a1 (a2 ) , b1 (b2 ) , a1 , b1 a2 , b2 g , Q ↦ Q ↦ via the identifications ai a, bi b. Under this change of presentation the word vm(XQ) gets identified with the word wm({a, b}) under this identification. Thus, AreaQ(vm(XQ)) ≍ 2 δQ(m)≍ m . → It follows that the homomorphism φ ∶ G Q as defined at the beginning of this section, ν as defined in (5.1) and hm satisfy all conditions of Corollary 4.3. We obtain 2  that ShmSK2 ≽ δQ(m) ≍ m . Remark 5.3. Note that in fact the same argument works for any sequence of elements (2) gm which is contained in an undistorted free subgroup H ≤ Γ2 , has reduced length asymptotically equivalent to m and can be represented by words vm(X ) satisfying the conditions of Proposition 4.2. Thus the conclusions of Lemma 5.2 apply to this more 1 general class of elements in K2 ∩ ‰Γ( ) × {1}Ž.

The rest of the proof of Theorem 5.1 will require a suitable decomposition of K3 to which we can apply the following consequence of [25, Theorem 6.1].

Theorem 5.4. Let Λ = G1∗H G2 be finitely presented with H a proper subgroup of each Gi, and H = ⟨B⟩, G1 = ⟨A1⟩ and G2 = ⟨A2⟩, where B, A1 and A2 are finite generating sets. Let P = ⟨A1, A2 S R⟩ be a finite presentation for Λ. Then there is a constant C = C(P, B) > 0 such that the following holds: 10

Given elements h ∈ H, g1 ∈ G1 ∖ H and g2 ∈ G2 ∖ H with [g1, h] = [g2, h] = 1, and words w = w(A1) representing h and ui = ui(Ai) representing gi, i = 1, 2, then Area w, u u n C n dist 1, h P ([ ( 1 2) ]) ≥ ⋅ ⋅ Cay(H,B)( )

Note that [25, Theorem 6.1] shows that for an explicit presentation of Λ of the form ⟨A1, A2, B S R⟩ one can choose C = 2. It is straight-forward to see that Theorem 5.4 follows from this result, using that Area and dist are equivalent up to multiplicative P Cay(H,B) constants under change of presentation and generating set. (1) (2) (3) Projection of the subgroup K3 ≤ Γ2 × Γ2 × Γ2 onto the third factor induces a short exact sequence p3 → → → (3) → 1 K2 K3 Γ2 1 (1) (2) → 2 with K2 = ker ŠΓ2 × Γ2 Z . The homomorphism

(3) → (1, ν, idΓ2 ) ∶ Γ2 K3

(3) provides a splitting of this sequence. Hence, it follows that K3 ≅ K2 ⋊ Γ2 is a semidirect product. (3) (3) (3) (3) (3) Consider the decomposition Γ2 Free a1 , b1 (3) (3) (3) (3) Free a2 , b2 = ( ) ∗a1 ,b1 =b2 ,a2  ( ) into an amalgamated product over Z. Its lift under p3 provides a decomposition of K3 as amalgamated free product

(3) (3) (3) (3) K3 ≅ ŠK2 ⋊ Free(a1 , b1 ) ∗H ŠK2 ⋊ Free(a2 , b2 ) , with −1 (3) (3) (3) (3) (3) (3) H = p3 Šba1 , b1 g = K2 ⋊ ba1 , b1 g ≅ K2 × ba1 , b1 g ,

(3) (3) (3) (3) where the last identity follows from the fact that Š1, ν(a1 ),ν(b1 ) , a1 , b1  K2 = (3) (3) Š1, 1, a1 , b1  K2 as cosets of K2 ⊴ H. (3) (3) To simplify notation we will now write K2 × Z for K2 × ba1 , b1 g.

1 Lemma 5.5. Z ( ) Z Let h ∈ (K2 × ) ∩ ŠΓ2 × {1} × {1}. Then ShSK2× ≍ ShSK2 .

(3) (3) Proof. Let Y be a generating set for K2 and let z = Š1, 1, a1 , b1 . Then Y+ =Y ∪ {z} is a generating set for K2 × Z. It is clear that with respect to this generating set we have Z ShSK2× ≤ ShSK2 . Conversely, let ω(Y, z) be a word of minimal length in K2 ×Z representing h in K2 ×Z. Then ω(Y, z) = ω(Y, 1)ω(1, z) and in particular ω(1, z) represents the trivial element in Z, while ω(Y, 1) is a word in Y representing h. Hence, we obtain

Z ShSK2× = l (ω(Y, z)) = l (ω(Y, 1)ω(1, z)) ≥ l (ω(Y, 1)) ≥ ShSK2 . 

As a consequence of Lemma 5.5 and Lemma 5.2, we obtain

(1) m (1) m Corollary 5.6. For m ≥ 1, the element hm = Š(a1 ) , (b2 )  , 1, 1 ∈ K2 × Z satisfies 2 Z ShmSK2× ≽ m . 11

(3) (3) (3) (3) Proof of Theorem 5.1. Let G1 = K2⋊Free(a1 , b1 ) and G2 = K2⋊Free(a2 , b2 ). Extend 3 3 1 3 1 2 ( ) ( ) ( ) ( ) −1 ( ) ( ) −1 the set šŠ1,ν(a1 ), a1  , Ša1 , 1, (a1 )  , Šb2 , (b2 ) , 1Ÿ ⊂ G1 to a finite generating (3) (3) set A1 of G1 and the set šŠ1,ν(a2 ), a2 Ÿ ⊂ G2 to a finite generating set A2 of G2 and choose any finite generating set B for H. Let further P = ⟨A1, A2 S R⟩ be a finite presentation for K3. Choose elements represented by words

(3) (3) (3) (3) g1 = u1(A1) = Š1,ν(a1 ), a1  ∈ ŠK2 ⋊ Free(a1 , b1 ) ∖ H and (3) (3) (3) (3) g2 = u2(A2) = Š1,ν(a2 ), a2  ∈ ŠK2 ⋊ Free(a2 , b2 ) ∖ H.

(1) Since hm ∈ K3 ∩ Γ2 , we have [g1, hm] = [g2, hm] = 1. The element hm is represented by the word m m (1) (3) −1 (1) (2) −1 wm(A1) = Ša1 , 1, (a1 )  , Šb2 , (b2 ) , 1  of length l(wm)= 4m with respect to the generating set A1. Thus, by Theorem 5.4 there is C = C(P, B)> 0 such that Area w , u u m C m dist 1, h . P ([ m ( 1 2) ]) ≥ ⋅ ⋅ Cay(H,B)( m) m m The word [wm, (u1u2) ] has length l([wm, (u1u2) ]) = 12m in the alphabet A1 ∪A2. Hence, it follows from Corollary 5.6 that

m 3 Z δK3 (m) ≽ AreaP ([wm, (u1u2) ]) ≽ CmShmSK2× ≽ m . This completes the proof of Theorem 5.1. 

Note that K3 is a K¨ahler group of type F2, not of type F3 (see [23]), which contains the subgroup K2 of type F1, not of type F2. In particular, K3 is not coherent. Hence, we can summarize the main properties of K3 as follows.

Theorem 5.7. The group K3 is a non-coherent K¨ahler group of type F2, not of type F3. 3 The Dehn function of K3 is bounded below by δK3 (n) ≽ n . As a consequence we obtain:

Proof of Corollary 1.2. Dimca, Papadima and Suciu obtain the group K3 as fundamen- tal group of the (compact) smooth generic fibre H of a surjective holomorphic map (1) (2) (3) → (1) (2) (3) f ∶ S2 × S2 × S2 E for E an elliptic curve. In particular, H ⊂ S2 × S2 × S2 (1) (2) (3) is an embedded complex submanifold. The manifold S2 × S2 × S2 admits a metric with non-positive holomorphic bisectional curvature, since it is a product of closed Rie- mann surfaces. Since holomorphic bisectional curvature is non-increasing when passing to complex submanifolds [29, Section 4], it follows that H can be endowed with a metric of non-positive holomorphic bisectional curvature. By Theorem 5.7, K3 does not admit a quadratic isoperimetric function and the result follows. 

Note that Corollary 1.2 is in contrast to the case of non-positive (Riemannian) sec- tional curvature, as the latter implies that the fundamental group admits a quadratic isoperimetric function. 12

6. An upper bound

For a group G with finite presentation G = ⟨X S R⟩, an area-radius pair (α, ρ) is a pair N → R N → R of functions α ∶ >0 and ρ ∶ >0 such that for every null-homotopic word w(X) k 1 1 with l w X n, there is a free equality w X ui X r± ui X − with k α n ( ( )) ≤ ( ) = ∏j=1 ( ) i ( ( )) ≤ ( ) and max {l(ui(X)) S 1 ≤ i ≤ k} ≤ ρ(n). Note that area-radius pairs are independent of the choice of presentation (up to equivalence). For further details on area-radius pairs we refer the reader to [24, Section 3]. An upper bound on the Dehn function of K3 can be obtained as a consequence of the following theorem by Dison:

Theorem 6.1 ([24, Theorem 11.15]). For r ≥ 3, let G1,...,Gr be finitely presented groups with area-radius pairs (αi, ρi), for 1 ≤ i ≤ r, and let A be an abelian group of rank m = 2 dim (A ⊗Z Q). Define α(n) = max ™n , αi(n), 1 ≤ i ≤ rž, ρ(n) = max {n, ρi(n), 1 ≤ i ≤ r}. → If φ ∶ G1 ×⋅⋅⋅× Gr A is a homomorphism such that the restriction of φ to each factor 2m Gi is surjective, then ρ α is an isoperimetric function for kerφ.

Lemma 6.2. Let G = ⟨X S R⟩ be a finitely presented group and let δG be the Dehn function of G. Then there is C > 0 such that (δG(n),C ⋅ δG(n) + n) defines an area-radius pair for G.

Proof. It is not hard to see that, given a van Kampen diagram of area k = Area(w(X)) for a word w(X), then w(X) is freely equal to a word of the form k ±1 −1 w(X) = M uj(X)rj (uj(X)) j=1 with l(ui(X)) ≤ C ⋅ k + l(w(X)), where C = max {l(r(X)) S r ∈ R}.  Hence, (δG(n), CδG(n) + n) is an area-radius pair for G. As an easy consequence we obtain an upper bound on the Dehn function of the group K3.

6 Corollary 6.3. The Dehn function of K3 satisfies δK3 (n) ≼ n . (1) (2) (3) → 2 Proof. The group K3 is the kernel of the homomorphism θ3 ∶ Γ2 ×Γ2 ×Γ2 Z which (i) is surjective on factors. Since surface groups are hyperbolic the groups Γ2 have linear Dehn function δ (i) n n. By Lemma 6.2 they admit area-radius pairs αi, ρi with G2 ( ) = ( ) 4 6  αi n ρi n δ (i) n n. Theorem 6.1 implies that δK3 n ρ α n . ( ) ≍ ( ) ≍ Γ2 ( ) ≍ ( ) ≼ ⋅ ≍ Note that Theorem 6.1 was used in a very similar way in [24, Corollary 12.6] to show that subdirect products of r limit groups of type Fr−1 have polynomial Dehn function and in [24, Proposition 13.3(2)] to show that the examples in [25] have Dehn function bounded above by n6. Proof of Theorem 1.1. This is now an immediate consequence of Theorem 5.7 and Corol- lary 6.3. 

7. Questions This work begs intriguing questions and we want to finish by listing some of them. One may first ask whether our result can be extended as follows. 13

Question 2. For every integer k, is there k′ ≥ k and a K¨ahler group G with Dehn function k k′ satisfying n ≼ δG(n) ≼ n ? Note that the same reasoning as in [24, Proposition 13.3(3)] and Section 6 can be applied to see that the groups constructed in [23], [5] and [38] must all have Dehn function bounded above by n6. Hence, these examples can not be used to obtain a positive answer to this question. However, these arguments do not apply to many of the groups constructed very recently by the first author from maps onto higher-dimensional tori [37]. Thus, in the light of Theorem 1.1, one could try to search for groups with larger Dehn function among these examples. Recall that the gap between 1 and 2 in Theorem 2.1(2) is the only gap in the isoperi- metric spectrum, that is, for every α ∈ [2, ∞) and ǫ > 0 there is a group G with Dehn β function n and Sα − βS < ǫ [8]. It is thus natural to ask if the same is true in the class of K¨ahler groups.

Question 3. What is the isoperimetric spectrum of Dehn functions of K¨ahler groups? Does it contain any gaps on [2, ∞)? We do not know the exact asymptotic of the Dehn function of the group of Theorem 1.1, and in particular we do not know whether there exists a single point in the isoperimetric spectrum of Dehn functions of K¨ahler groups that is different from 1 or 2. We might wonder whether there exist K¨ahler groups with arbitrary large Dehn function. For instance

Question 4. Do all K¨ahler groups have solvable word problem?

Recall that the word problem is solvable if and only if the Dehn function is a recursive function. At this time even the following question is open.

Question 5. Do all K¨ahler groups admit an exponential isoperimetric function?

Finally, an interesting though mysterious class of K¨ahler groups are those which are nilpotent. The only non-trivial known examples are the higher dimensional Heisenberg groups mentioned in Section 3 (which have quadratic Dehn function).

Question 6. Do all nilpotent K¨ahler groups admit a quadratic Dehn function?

More generally, we do not know whether nilpotent groups whose Malcev algebras are quadratically presented admit a quadratic Dehn function, although in the introduction of the first arXiv version of his article [51], Young suggests a strong link between these two conditions (see also [46, Section 6.2]).

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Laboratoire de Mathematiques´ d’Orsay, Univ. Paris-Sud, CNRS, Universite´ Paris-Saclay, 91405 Orsay, France E-mail address: [email protected]

Institut de Mathematiques´ de Jussieu-PRG, Universite´ Paris-Diderot, Case 7012, 75205 Paris Cedex 13, France E-mail address: [email protected]