THE UNIVERSALITY OF HUGHES-FREE DIVISION RINGS

Let E ∗G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E ∗G-isomorphism, there exists at most one Hughes-free division E ∗ G-ring. The existence of a Hughes-free division E ∗G-ring DE∗G is an open question, but it exists, for example, if G is amenable or G is bi-orderable. We study, whether DE∗G is universal division ring of fractions in these cases. In Appendix we give a description of DE[G] when G is a RFRS group, generalizing a result of Kielak.


Introduction
A division R-ring φ : R → D is called epic if φ(R) generates D as a division ring. Each division R-ring D induces a Sylvester matrix rank function rk D on R. Given a ring R, Cohn introduced the notion of universal division R-ring (see, for example, [4,Section 7.2]). In the language of Sylvester rank functions, an epic division R-ring D is universal if for every division R-ring E, rk D ≥ rk E . By a result of Cohn [3,Theorem 4.4.1], it follows that the universal division R-ring is unique up to R-isomorphism. The universal division R-ring D is called universal division ring of fractions of R if D is epic and rk D is faithful (that is R is embedded in D).
If R is a commutative domain, then the field of fractions Q(R) is the universal division R-ring. The situation is much more complicated in the non-commutative setting. For example, Passman [25] gave an example of a Noetherian domain which does not have a universal division ring of fractions. Moreover, we show in Proposition 4.1 that the group algebra Q[H] does not have a universal division ring of fractions if H is not locally indicable. In this paper we want to study whether a group algebra or, more generally, a crossed product E * G, where E is a division ring, has a universal division ring of fractions. Thus, from the previous observation it is natural to consider the case of group algebras and crossed products E * G where G is locally indicable.
Let E be a division ring and G a locally indicable group. Hughes [12] introduced a condition on an epic division E * G-rings and showed that up to E * G-isomorphism, there exists at most one epic division E * G-ring satisfying this condition. We call this division ring, the Hughes-free division E * G-ring and denote it by D E * G . For simplicity, in this paper the Sylvester matrix rank function rk D E * G is denoted by rk E * G . We say that a locally indicable group G is Hughes-free embeddable if E * G has a Hughes-free division ring for every division ring E and every crossed product E * G.
The existence of a Hughes-free division E * G-ring is known for several families of locally indicable groups. In the case of amenable locally-indicable groups G, D E * G = Q(E * G) is the classical ring of fractions of E * G, and in the case of biorderable groups G, D E * G is constructed using the Malcev-Neumann construction [21,24] (see [9]). The existence of D K [G] is also known for group algebras K[G], where K is of characteristic 0 and G is an arbitrary locally indicable group [16].
In [16,Theorem 8.1] it is shown that if there exists a universal epic division E * G-ring and a Hughes-free division E * G-ring, they should be isomorphic as E * G-rings. Following Sánchez (see [26,Definition 6.18]), we say that a locally indicable group G is a Lewin group if it is Hughes-free embeddable and for all possible crossed products E * G, where E is a division ring, D E * G is universal (in Subsection 3. 3 we will see that this definition is equivalent to the Sánchez one). We conjecture that all locally indicable groups are Lewin. We want to notice that at this moment it is also an open problem of whether the universal division E * G-ring of fractions (if exists) should be Hughes-free.
In this paper we study part (B) of the conjecture in some cases where part (A) is known. Using Theorem 3.7 we can show that Conjecture 1 is valid for the following locally indicable groups. Theorem 1.1. Locally indicable amenable groups, residually-(torsion-free nilpotent) groups and free-by-cyclic groups are Lewin groups.
In the case of group algebras we can prove a stronger result. The metric space G n of marked n-generated groups consists of pairs (G; S), where G is a group and S is an ordered generating set of G of cardinality n. Such pairs are in 1-to-1 correspondence with epimorphisms F n → G, where F n is the free group of rank n, and thus the set G n can be identified with the set of all normal subgroups of F = F n . The distance between M 1 and M 2 is defined by where B k (1 F ) denotes the closed ball of radius k and center 1 F . We say that a sequence of n-generated groups {G i } i∈N converges to an ngenerated group G if (G i ; S i ) ∈ G n converge to (G; S) ∈ G n for some generating sets S i of G i (i ∈ N) and S of G, respectively.
The corollary can be applied to RFRS groups, because they are residually poly-Z. In Appendix we give a description of D E[G] when G is a RFRS group that generalizes a result of Kielak [19].
Let us consider now the case of group algebras K[G] where K is a subfield of C and G is locally indicable. In this case it was shown in [16] that the division closure D(K[G], U(G)) of K [G] in the algebra of affiliated operators U(G) is a Hughesfree division K[G]-ring. We denote by rk G the von Neumann rank function (its definition is recalled in Subsection 2.6), and by rk {1} the Sylvester matrix rank function on Q[G] induced by the homomorphism Q[G] → Q that sends all the elements of G to 1 (in the previous notation rk {1} is rk Q ). In view of Conjecture 1, it is natural to ask for which groups G, rk G ≥ rk {1} . It follows from [27,Proposition 1.9] that if a group G satisfies the condition rk G ≥ rk {1} , then G is locally indicable. Thus, we propose also a weak version of Conjecture 1. From the discussion in the paragraph before the conjecture, we conclude that Corollary 1.3 has the following consecuence. Using a description of the division ring D Q[G] for RFRS groups G, Kielak [19] obtained a criterion of virtually fiberness of RFRS groups that generalizes a celebrated result of I. Agol [1] showing that an irreducible 3-manifold, whose fundamental group is RFRS, is virtually fibered. Since RFRS groups are residually poly-Z, we can apply Corollary 1.4, and together with Kielak's criterion we obtain the following corollary. Corollary 1.5. Let G be a finitely generated group which is virtually RFRS. Then the following are equivalent.
(1) G is virtually fibered, in the sense that it admits a virtual map onto Z with finitely generated kernel. (2) G admits a virtual map onto Z whose kernel has finite first Betti number.
Our next result is another consequence of Corollary 1.4 that generalizes a result of Wise from [29, Theorem 1.3], Corollary 1.6. Let X be a compact CW -complex with π 1 X non-trivial residually-(locally indicable and amenable) group. Then The paper is structured as follows. We introduce the basic notions in Section 2. In Section 3, we prove Theorem 1.1, Theorem 1.2 and Corollary 1.3. In Section 4 we study the consequences of the condition rk G ≥ rk {1} and, in particular, we prove Corollary 1.5 and Corollary 1.6. In Appendix we describe the division ring D E[G] when G is RFRS and E is a division ring.
(SEV-2015-0554). I would like to thank Dawid Kielak for useful suggestions and comments.

Preliminaries
2.1. Notation and definitions. All rings in this paper are unitary and ring homomorphisms send the identity element to the identity element. By a module we will mean a left module. Let G be a group with trivial element e. We say that a ring R is G-graded if R is equal to the direct sum ⊕ g∈G R g and R g R h ⊆ R gh for all g and h in G. If for each g ∈ G, R g contains an invertible element u g , then we say that R is a crossed product of R e and G and we will write R = S * G if R e = S. In the following if H is a subgroup of G, S * H will denote the subring of R generated by S and {u h : h ∈ H}.
A ring R may have several different G-gradings. It will be always clear from the context what G-grading we use. However, in some situations the grading is unique. Assume that R ∼ = E * G, where E is a division ring and G is locally indicable, then by [10], the invertible elements U (R) of R are g∈G R g \ {0}. Hence R e is the maximal subring in U (R) ∪ {0} and G ∼ = U (R)/(R e \ {0}). Thus, R has a unique grading with R e is a division ring and G is locally indicable.
An R-ring is a pair (S, φ) where φ : R → S is a homomorphism. We will often omit φ if it is clear from the context.

Ordered groups. A total order on a group
Let be a left-invariant order on a group G. A subgroup H is called convex if H contains every element g lying between any two elements of H (h 1 g h 2 with h 1 , h 2 ∈ H). We say that is Conradian if for all elements f, g 1, there exists a natural number n such that f g n g. In fact, one may actually take n = 2 ([6, Proposition 3.2.1]). Recall that a group G is locally indicable if every finitely generated non-trivial subgroup of G has an infinite cyclic quotient. A useful characterization of locally indicable groups says that they are the groups admitting a Conradian order ( [5]). We will need the following important property of a Conradian order.
Proposition 2.1. [6, Corollary 3.2.28] Let (G, ) be a group with a Conradian order and let N be a maximal convex subgroup of G. Then there exists an order preserving homomorphism φ : G → R such that N = ker φ.

2.3.
Hughes-free division rings. Let E be a division ring and G a locally indicable group. Let ϕ : E * G → D be a homomorphism from E * G to a division ring D. We say that a division E * G-ring (D, ϕ) is Hughes-free if (1) D is the division closure of ϕ(E * G) (D is epic).
(2) For every non-trivial finitely generated subgroup H of G, a normal subgroup N of H with H/N ∼ = Z, and h 1 , . . . , h n ∈ H in distinct cosets of N , the sum is the division closure of ϕ(E * N ) in D.) Hughes [12] (see also [7]) showed that up to E * G-isomorphism there exists at most one Hughes-free division ring. We denote it by D E * G . The uniqueness of Hughesfree division rings implies that for every subgroup H of G, D H,D E * G is Hughes-free as a division E * H-ring.
Gräter showed in [9,Corollary 8.3] that D E * G (if it exists) is strongly Hughesfree, that it satisfies the following additional conition: (2') For every non-trivial subgroup H of G, a normal subgroup N of H and h 1 , . . . , h n ∈ H in distinct cosets of N , the sum D N,D E * G ϕ(u h1 ) + . . . + D N,D E * G ϕ(u hn ) is direct. In particular, this implies the following result that we will use often without mentioning it explicitly. Proposition 2.2. Let G be a locally indicable group, N a normal subgroup of G and E a division ring. Assume that for a crossed product E * G, D E * G exists. Then the ring R generated by D N,D E * G and G has structure of a crossed product D E * N * (G/N ). In particular, (1) if N is of finite index in G, then D E * G = D E * N * (G/N ) and (2) if G/N is abelian, D E * G is isomorphic to the classical Ore ring of fractions of D E * N * (G/N ).

2.4.
Free division E * G-ring of fractions. Let G be group with a Conradian left-invariant order (so, G is locally indicable). Let E be a division ring. Let ϕ : E * G → D be a homomorphism from a crossed product E * G to a division ring D. We say that a division E * G-ring (D, ϕ) is free with respect to if (1) D is the division closure of ϕ(E * G).
(2) For every subgroup H of G, and the maximal convex subgroup N of H (which is normal by Proposition 2.1), and h 1 , . . . , h n ∈ H in distinct cosets of N , the sum D N,D ϕ(u h1 ) + . . . + D N,D ϕ(u hn ) is direct. This notion was introduced by Gräter in [9]. Remark 2.3. Notice that in part (2) of the definition, we also can assume that H is finitely generated. Indeed, assume (2) holds for finitely generated subgroups, but for some H and h 1 , . . . , h n , there are d 1 , . . . , d n ∈ D N,D , not all equal to zero, such that d 1 ϕ(u h1 ) + . . . + d n (u hn ) = 0. Then we can find a finitely generated subgroup of N of N such that d 1 , . . . , d n ∈ D N ,D . Let H be the subgroup of G generated by h 1 , . . . , h n and N . Since n ≥ 2, N ∩ H is the maximal convex subgroup of H . This contradicts our assumption that (2) holds for H .
Gräter proved the following result. and let E be a division ring. A division E * G-ring is free with respect to if and only if it is Hughes-free (and so, it is E * G-isomorphic to D E * G ).

2.5.
Sylvester matrix rank functions. Let R be a ring. A Sylvester matrix rank function rk on R is a function that assigns a non-negative real number to each matrix over R and satisfies the following conditions. We denote by P(R) the set of Sylvester matrix rank functions on R, which is a compact convex subset of the space of functions on matrices over R. If φ : F 1 → F 2 is an R-homomorphism between two free finitely generated R-modules F 1 and F 2 , then rk(φ) is rk(A) where A is the matrix associated with φ with respect to some R-bases of F 1 and F 2 . It is clear that rk(φ) does not depend on the choice of the bases. A useful observation is that a ring homomorphism ϕ : R → S induces a continuous map ϕ : P(S) → P(R), i.e., we can pull back any rank function rk on S to a rank function ϕ (rk) on R by just defining ϕ (rk)(A) = rk(ϕ(A)) for every matrix A over R. We will often abuse the notation and write rk instead of ϕ (rk) when it is clear that we speak about the rank function on R.
A division ring D has a unique Sylvester matrix rank function which we denote by rk D . If a Sylvester matrix rank function rk on R takes only integer values, then by a result of P. Malcolmson [22] there are a division ring D and a homomorphism ϕ : R → D such that rk = ϕ (rk D ). Moreover, if D is equal to the division closure of ϕ(R) (D is an epic division R-ring), then ϕ : R → D is defined uniquely up to isomorphisms of R-rings. We denote the set of integer-valued rank functions on a ring R by P div (R). In the following, if a rank function on R is induced by a homomorphism to D we will also use rk D to denote this rank function (in this case the homomorphism will be clear from the context).
Given two Sylvester matrix rank functions on R, rk 1 and rk 2 , we will write rk 1 ≤ rk 2 if for any matrix A over R, rk 1 (A) ≤ rk 2 (A). In the case where both functions are integer-valued and come from homomorphisms ϕ i : R → D i (i = 1, 2) from R to epic division rings D 1 and D 2 , the condition rk D1 ≤ rk D2 is equivalent to the existence of a specialization from D 2 to D 1 in the sense of P. M. Cohn ([3, Subsection 4.1]). We say that an epic division R-ring D is universal if for every epic division R-ring E, rk D ≥ rk E .
An alternative way to introduce Sylvester rank functions is via Sylvester module rank functions. A Sylvester module rank function dim on R is a function that assigns a non-negative real number to each finitely presented R-module and satisfies the following conditions.
There exists a natural bijection between Sylvester matrix and module rank functions over a ring. Given a Sylvester matrix rank function rk on R and a finitely presented R-module M ∼ = R n /R m A (A is a matrix over R), we define the corresponding Sylvester module rank function dim by means of dim(M ) = n−rk(A). If a Sylvester matrix rank function rk D comes from a division R-ring D, then the corresponding Sylvester module rank function will be denoted by dim D . Then D is the universal epic division R-ring if and only if for every epic division R-ring E and every finitely presented By a recent result of Li [20], any Sylvester module rank function on R can be extended to arbitrary modules over R. In the case of an integer-valued Sylvester module rank function dim D and an R-module M we simply have dim D (M ) = dim D (D ⊗ R M ).
2.6. Von Neumann rank function. Consider first the case where G is countable. Then G acts by left and right multiplication on the separable Hilbert space l 2 (G). A finitely generated Hilbert G-module is a closed subspace V ≤ l 2 (G) n , invariant under the left action of G. We denote by proj V : l 2 (G) n → l 2 (G) n the orthogonal projection onto V and we define where 1 i is the element of l 2 (G) n having 1 in the ith entry and 0 in the rest of the entries.
Then the group elements that appear in A are contained in a finitely generated group H. We will put rk G (A) = rk H (A). One easily checks that the value rk H (A) does not depend on the subgroup H.
Another obvious Sylvester matrix rank function on G arises from the trivial homomorphism G → {1} and it is defined as where A is the matrix over C obtained from A by sending all the elements of G to 1.
2.7. The natural extension. Let R = E * G be a crossed product of a division ring E and a group G. Let N be a normal subgroup of G such that G/N is amenable. Consider a transversal X of N in G. Since G/N is amenable there are finite subsets X k of X such that {N X k /N } is a Følner sequence in G/N with respect to the right action. Put X k = N X k .
Let rk be a Sylvester rank function on E * N and assume that rk is invariant under conjugation by the elements {u g } g∈G . Observe that if rk = rk E for some epic division E * N -ring E, then the conjugation of E * N by any u g (g ∈ G) can be extended to a unique automorphism of E. Thus one can consider the crossed product E * G/N containing E * G.
Let A ∈ Mat n×m (R) and let S be the union of supports of the entries of A. For any subset T of G we denote R T = ⊕ t∈T R t . Let φ k : (R X k ) n → (R X k S ) m be the homomorphism of finitely generated free E * N -modules induced by the right multiplication by A. Let ω be a non-principal ultrafilter on N. Then we put (1) rk Then rk ω is a Sylvester rank function on R. The rank function rk ω has been already studied previously in different situations (see [28,15,16,18]). In [18] it is shown that rk ω does not depend on ω. Therefore in the following we denote rk ω by rk.
The Sylvester rank function rk is called the natural extension of rk. We describe now the cases that appear in this paper.
Proposition 2.5. Let G be a group with a normal subgroup N such that G/N is amenable. Let E be a division ring, and assume the previous notation. Then the following holds.
(1) Assume that N and G/N are locally indicable and rk = rk E for some epic division E * N -ring E. Then rk coincides with rk Q(E * (G/N )) , where Q(E * (G/N )) denotes the classical Ore ring of fractions of E * (G/N ). Proof.
(1) We can extend rk to a Sylvester matrix rank function on E * (G/N ) (which we denote also by rk) using the formula (1). Since G/N is locally indicable, the ring E * (G/N ) is a domain. Thus, by the definition of rk, rk(a) = 1 for every 0 = a ∈ E * (G/N ). Hence, applying [15, Proposition 5.2], we obtain that rk = rk Q(E * (G/N )) . The statements (2) and (3)  3. On the universality of D E * G

3.1.
A general criterion of universality. In this subsection we present a general criterion of universality of a division R-ring. The proof of the following lemma is immediate.
Lemma 3.1. Let R be a ring and E a division R-ring. Let M be a finitely generated left R-module. Then the following are equivalent.
The following proposition tells us that in order to check universality of a division R-ring D it is enough to understand the structure of its finitely generated R-submodules.
This proves the "only if" part of the proposition. Now, consider the "if" part. We want to show that for every finitely generated left R-module M and every division R-ring E, dim E (M ) ≥ dim D (M ). We will do it by induction on dim D (M ).
Let M be the image of the natural R-homomorphism α : We have also that dim E (M ) ≤ dim E (M ). Thus, without loss of generality, we can assume that α is injective.
. This gives us the base of induction.
Assume that the claim Applying the inductive assumption we obtain that 3.2. The universality of D E * G in the amenable case. Let E be a division ring and G a locally indicable group. Proposition 3.2 indicates that in order to prove the universality we have to understand the structure of finitely generated E * Gsubmodules of D E * G . If G is amenable, they are isomorphic to finitely generated left ideals of E * G. The following result shows that in the latter case the condition of Proposition 3.2 holds. Proposition 3.3. Let R = E * G be a crossed product of a division ring E and a locally indicable group G. Then for every non-trivial finitely generated left ideal L of R and every division R-ring E, dim E (L) > 0.
Proof. We denote by R g the gth component of R and let u g be an invertible element of R g . Then R e ∼ = E. For any element r = g∈G r g ∈ R (r g ∈ R g ) denote by supp(r) the elements g ∈ G for which r g = 0 and put l(r) to be equal to the number of non-trivial elements in supp(r). Thus, l(r) = 0 means that r ∈ R e . For a non-trivial finitely generated left ideal L of R we put l(L) = min{l(r 1 ) + . . . + l(r s ) : L = Rr 1 + . . . + Rr s }.
Observe that if a set of generators {r 1 , . . . , r s } of L satisfies the equality l(L) = l(r 1 ) + . . . l(r s ), then for each i, l(r i ) = | supp(r i )| − 1. (If not, we can change r i by u −1 g r i with g ∈ supp(r i ) and obtain a contradiction.) Moreover, if all r i are non-trivial and L = R, then s ≤ l(L). Now, we define s(L) = max{s : L = Rr 1 +. . .+Rr s , l(L) = l(r 1 )+. . .+l(r s ) and r i are non-trivial}.
We will prove the proposition by induction on l(L). If l(L) = 0, then L = R and we are done. Now assume that the proposition holds if l(L) ≤ n − 1, and consider the case l(L) = n ≥ 1.
We will proceed by inverse induction on s(L). Observe that there is no L such that s(L) ≥ l(L) + 1, so there is nothing to prove in this case. Assume that we can prove the proposition if l(L) = n and s(L) ≥ k + 1 , and consider the case l(L) = n and s(L) = k.
Let r 1 , . . . r k be a set of non-zero generators of L such that n = l(r 1 ) + . . . l(r k ). Let H be the group generated by ∪ k i=1 supp(r i ). Since G is locally indicable there exists a surjective α : H → Z. Let N = ker α and t ∈ H such that t N = H. We write Thus, we obtain that either l(L ) < l(L) or l(L ) = l(L) and s(L ) > s(L).
Hence we can apply the inductive hypothesis and obtain that rk E (L ) > 0. Thus , where τ is conjugation by u t . Then E has a natural S-ring structure. We denote by dim E the corresponding Sylvester module rank function on S. By Proposition 2.5(1), rk E is equal to the natural extension of the restriction of rk E on E * N .
Let L 0 and L 0 be the left S-submodules of D generated by {r i } and {r ij } respectively. We have that L 0 ≤ L 0 . Every element m of L 0 can be written in a unique way as Let dim E be the Sylvester module rank function associated to the division S-ring E. Since the restrictions of rk E and rk E on E * N coincide [16,Lemma 8.3] implies that rk E ≤ rk E as Sylvester matrix rank functions on E * H, and so and we are done.
Corollary 3.4. Let G be an amenable locally indicable group and let E be a division ring. Then D E * G is the universal division ring of fractions of E * G.
Proof. Observe that E * G satisfies the left Ore condition and so D E * G is isomorphic as E * G-ring to the classical ring of fractions Q(E * G). Since any finitely generated left submodule of Q(E * G) is isomorphic to a left ideal of E * G, Proposition 3.2 and Proposition 3.3 imply the desired result.
We remark that Corollary 3.4 can be also proved using arguments similar to the ones used in the proof of [11, Lemma 2.1]. Also it is worth to be mentioned here that, by a result of D. Morris [23], a left orderable amenable group is always locally indicable.

3.3.
A criterion for a group to be Lewin. In this subsection we will show that in order to prove that a Hughes-free embeddable group G is Lewin, it is enough to consider only group algebras E[G]. As before, by rk E we denote the Sylvester matrix rank function on E[G] induced by the homomorphism E[G] → E that sends all the group elements from G to 1. If for a crossed product E * G, the space P div (E * G) is not empty, then E * G has the Hughes-free division ring D E * G and, moreover, D E * G is universal.
Proof. First let us show that D E * G exists. Let φ : E * G → E be a division E * Gring. Write R = E * G = ⊕ g∈G R g . We fix an invertible element u g ∈ R g for each g ∈ G. For every g 1 , g 2 ∈ G we define Observe that E is a E * G-bimodule. This allows us to convert the E-spaceR = ⊕ g∈G Ev g into a ring by putting v g a = (φ(u g )aφ(u −1 g ))v g and v g v h = φ(α(g, h))v gh , g, h ∈ G, a ∈ E. Clearly the ringR has a structure of a crossed productR = E * G. Define the map Then φ is a homomorphism.
For each g ∈ G we put w g = φ(u −1 g )v g ∈ E * G. Then w g commutes with the elements from E and for every g, h ∈ G, In particular D E * G , and so, D E * G exist and φ # (rk D E * G ) is equal to rk D E * G . Now, we want to show that D E * G is universal: rk D E * G ≥ φ # (rk E ). Let ψ : E * G → E be the map that sends all w g to 1. Denote by rk E the Sylvester matrix rank function on E * G induced by ψ. By our assumptions, rk E ≤ rk D E * G . Now observe that φ = ψ • φ. Hence as Sylvester matrix rank functions on E * G. exists. Since G i are quotients of F , abusing notation, we will also refer to rk E[Gi] as a Sylvester matrix rank function on E[F ].
Let ω be an arbitrary non-principal ultrafilter on N. We put Observe that for every g ∈ M , rk(g − 1) = 0. Thus, rk is also a Sylvester matrix rank function on E[G]. We want to show that rk corresponds to the Sylvester matrix rank function of a Hughes-free division E * G-ring. This will prove Theorem 1.2.
For each i we fix a left-invariant Conradian order i on G i . Define an order on G by The definition does not depend on the choice of representatives, because for every m ∈ M , the set {i ∈ N : m ∈ M i } is in ω. It is also clear that is left-invariant and Conradian. In particular, this proves that G is locally indicable.
Denote by α j the canonical homomorphism F → G j and extend it to the homomorphism α j : . The rank function rk corresponds to the homomorphism . As we have observed before, for each m ∈ M , α(m − 1) = 0. Thus, D is a epic division E[G]-ring. We are going to show that D is free with respect to . For simplicity, in what follows, Let H be a finitely generated subgroup of G and let N be the maximal convex subgroup of H. Let h 1 , . . . , h n ∈ H be in distinct cosets of N . We want to show that α(h 1 ), . . . , α(h n ) are D N,Dω -linearly independent. Without loss of generality we will assume that H = G.
Let L j /M j be the maximal convex subgroup of G j with respect to j . By Proposition 2.1, since j is Conradian, there exists order-preserving homomorphism φ j : G j → R such that ker φ j = L j /M j . Without loss of generality we see φ j as an element of H 1 (F, R). We can multiply φ j by a scalar in such way that max s∈S |φ j (s)| = 1. Let φ = lim ω φ j ∈ H 1 (F, R) and L = ker φ. Observe that φ is non-trivial, M ≤ ker φ and φ is order-preserving with respect to if we consider it as a map G → R. In particular, N = L/M .
For each i choose f i ∈ F such that h i = f i M . By way of contradiction, assume that there are d 1 , . . . , d n ∈ D N,D such that Since h 1 , . . . , h n ∈ H belong to distinct cosets of N , all values φ(f 1 ), . . . , φ(f n ) are distinct. Let = min l =i |φ(f j ) − φ(f i )|. Since for all i, j, φ(f il ) = 0, we obtain that Thus, without loss of generality we assume that for every k ∈ N, |φ k (f il )| ≤ 4 for all i, l and |φ k ( ). Therefore, we can write Thus, we can assume that Since D k is free with respect to k , this implies that for all i, Since this holds for all k, d i = 0 for all i. This shows that D is free with respect to , and so it is Hughes-free by Proposition 2.4. This finishes the proof of Theorem 1.2.
Proof Corollary 1.3. Without loss of generality we may assume that G is finitely generated. Hence G is a limit of a collection of locally indicable amenable groups {G i }.  (1) If all finitely generated subgroups of G are Lewin, then G is also Lewin.
(2) Any subgroup of a Lewin group is also Lewin.
(3) G is Lewin if G has a normal Lewin subgroup N such that G/N is amenable and locally indicable. (4) Any limit in G n of Lewin groups which is Hughes-free embeddable is Lewin. (5) A finite direct product of Lewin groups is Lewin.
Proof. The first statement follows directly from the definition of Lewin groups and the second one from Corollary 3.6. Let us prove now part (3).
First observe that G is Hughes-free embeddable by [13] (see also [  The fourth statement follows from Proposition 3.5 and Theorem 1.2. Consider now the fifth claim. First let us prove that the direct product G = G 1 × G 2 of two Lewin groups G 1 and G 2 is again Lewin. By [13], G is Hughes-free embeddable. Let E be a division ring. Consider the natural homomorphisms . Since E is arbitrary, applying Proposition 3.5, we obtain that G is Lewin. The case of two groups implies that (5) holds for an arbitrary finite product of Lewin groups.

Universality of rk G
As we have already mentioned in Introduction, when G is locally indicable rk G = rk D C [G] . In this section we compare rk G with other natural Sylvester matrix rank functions on C[G].

4.1.
The condition rk G ≥ rk {1} . In this subsection we will see several consequences of the condition rk G ≥ rk {1} . Recall that rk {1} is an alternative expression for rk C that has appeared in the previous sections. We start with the following useful proposition. Let F be a free group generated by x 1 , . . . , x d . For each 1 ≤ i ≤ d, we write  In the next proposition we will show that the condition rk G ≥ rk {1} implies that rk G ≥ rk G for any amenable quotient G of G.  Proof. Without loss of generality we may assume that G is finitely generated. Then there exists a chain G = N 0 > N 1 > N 2 > · · · of normal subgroups of G such that G/N k is amenable and ∩N k = N . By [14,Theorem 1.3], By Proposition 4.2, rk G ≥ rk G/N k in P(K[G]) for every k. Hence rk G ≥ rk G/N holds as well.
We conjecture that the corollary holds without the condition that G/N is residually amenable.
Conjecture 3. Let G be a group and let K be a subfield of C. Assume that rk G ≥ rk {1} in P (K[G]). Then rk G ≥ rk G in P (K[G]) for any quotient G of G.

4.2.
Proof of Corollary 1.5. It is clear that part (1) of of Corollary 1.5 implies part (2). Kielak proved in [19] that in order to show (1), it is enough to prove that the first L 2 -Betti number of G is zero. Using Theorem 1.1, we will show that the condition (2) of Corollary 1.5 implies that the first L 2 -Betti number of G is zero.
First, let us recall the definition of RFRS groups. A group G is called residually finite rationally solvable or RFRS if there exists a chain G = H 0 > H 1 > . . . of finite index normal subgroups of G with trivial intersection such that H i+1 contains a normal subgroup K i+1 of H i satisfying that H i /K i+1 is torsion-free abelian. The following proposition implies that RFRS groups are residually poly-Z.
Proposition 4.4. Let G be a finitely generated group, and let be a chain of finite index normal subgroups of G with ∞ n=0 H n = 1. Suppose that for every n ≥ 0 there exists a subgroup K n+1 H n such that K n+1 ≤ H n+1 and H n /K n+1 is poly-Z. Then G is residually poly-Z. [17,Proposition 5.1]. The same proof works in our case. We include it for the convenience of the reader.

Proof. A pro-p version of this result is proved in
For n ≥ 1 let K n = g∈H/Hn−1 be the normal core of K n in G. Since the direct product of poly-Z-groups is poly-Z and a subgroup of a poly-Z group is poly-Z, the group H n−1 / K n is poly-Z as well. For every n ≥ 1 set and note that since ∞ n=0 H n = 1, this is a chain of subgroups that satisfies We shall argue, by induction on n, that G/L n is poly-Z. For n = 1 we have Once n ≥ 2 we have L n = L n−1 ∩ K n , and by induction G/L n−1 is poly-Z. Thus, since an extension of two poly-Z groups is poly-Z, it suffices to show that L n−1 /L n is poly-Z. Indeed, since K n−1 ≤ H n−1 , we have that L n−1 /L n = L n−1 /L n−1 ∩ K n ∼ = L n−1 K n / K n ≤ H n−1 / K n is poly-Z. Therefore, we conclude by recalling that a subgroup of a poly-Z group is poly-Z. Now let us prove that the condition (2) of Corollary 1.5 implies that the first L 2 -Betti number of G is zero. Let H be a subgroup of finite index such that there exists a normal subgroup N of H with H/N ∼ = Z and H 1 (N, Q) is finite-dimensional.
Assume that H has the following presentation. For

Denote by
is weakly exact. Therefore, the first L 2 -Betti number of H vanishes, and so the first L 2 -Betti number of G vanishes as well.

4.3.
Proof of Corollary 1.6. Consider the cellular chain complex of X Since G acts freely on X and X = X/G is of finite type, we obtain that -module of finite rank and the connected morphisms ∂ p are represented by multiplication by matrices A p over Z[G]. Hence we obtain the following equivalent representation of C( X): Therefore, if p ≥ 1 the pth Betti number of X and the pth L 2 -Betti number of X can be expressed in the following way.

Appendix: The universal division ring of fractions of group rings of division rings and RFRS groups
In this section G is assumed to be a finitely generated RFRS group and E is a division ring. By Proposition 4.4, G is residually poly-Z. Therefore, Corollary 1.3 implies that D E[G] exists and it is universal. In this section we will give an alternative description of D E[G] (see Theorem 5.10). Our proof follows essentially the argument of Kielak [19], where this description is done when E = Q.

Characters.
A character of G is a homomorphism from G to the additive group of real numbers R. The set of characters Hom(G, R) is denoted also by If H is a subgroup of finite index of G then the restriction map embeds H 1 (G; R) into H 1 (H; R). In what follows, we will often consider H 1 (G; R) as a subset of H 1 (H; R).
If H is a normal subgroup of G then G acts on H 1 (H; R): for φ ∈ H 1 (H; R) and g ∈ G, we denote by φ g the character that sends h ∈ H to φ(ghg −1 ).
Let G = H 0 > H 1 > H 2 > . . . be a chain of subgroups of G of finite index and n ≥ 0. For any U ⊂ H 1 (H n ; R) we denote We say that U is (G, {H i } i≥0 )-rich if U 0 contains all the irrational characters of G. When G and {H i } i≥0 are clear from the context, we will simply say that U is rich. (1) If U is rich in H 1 (H n ; R) and g ∈ G, then U g is also rich.
(2) The intersection of two rich subsets of H 1 (H n ; R) is again rich.
Proof. Claim (1) is clear. Let us show the second claim. First observe that if U and V are two open subsets of R k , then Consider y ∈ O(x), and let O(y) be an arbitrary neighborhood of y such that In particular, there exists z ∈ U ∩O(y). Recall that U is open. Consider an arbitrary Now let U and V be two rich subset of H 1 (H n ; R) and let W = U ∩ V . We put and similarly we define V k and W k . Then we have that W n = U n ∩ V n . Now, assume that we have proved that W k = U k ∩ V k for some k ≤ n. Then we obtain that In particular, W 0 contains contains all the irrational characters of G, and so, W is rich.
We will need the following criterion of richness.  (H n ; R). Assume that U is rich and all the irrational characters of U belong to V . Then V is also rich.
Then by the inverse induction on i, we obtain that all the irrational characters of U belong also to V i for n ≤ i ≤ k. Hence U ≤ V k . This clearly implies that V is rich.

5.2.
Novikov rings. Let S * G be a crossed product and let φ ∈ H 1 (G, R). Denote by φ a norm on S * G defined by Our convention is that 0 φ = 0. Let S * G φ be the completion of S * G with respect to the metric induced by the norm φ . The ring S * G φ is called the Novikov ring of R * G with respect to φ.
Let N = ker φ. Then φ is also a character of G/N and S * G φ is canonically isomorphic to (S * N ) * G/N φ . We will not distinguish between these two rings.
Any element of S * G φ can be represented in the following form ∞ i=1 a i g i , where a i ∈ R * N and {φ(g i )} i∈N is an increasing sequence tending to the infinity.
We would like to construct an environment, where we can calculate the inter- In order to do this, consider the following commutative diagram of of injective homomorphisms of rings. (4) where the maps are defined as follows.
Notice that D E Using this identifications, we obtain that α H,φ is the restriction of α G,φ . Therefore, in the following we will simply write α φ instead of α G,φ .
The map β G,φ can be defined as the the continuous (with respect to φ ) extension of the map It follows from the definitions that β H,φ is the restriction of β G,φ on E[H] φ . Thus, in the following we will simply write β φ instead of β G,φ .
For any subset S of H 1 (G, R) we put If φ ∈ H 1 (G, R), we will simply write D E In oder to prove (6), it is enough to show that for every φ ∈ S, x q ∈ D E[H],φ . We will do it in two steps. Put N = ker φ. In the first step we will assume that G = HN and in the second that N ≤ H. The combination of these two steps implies the general case of (6).
Thus, assume first that G = HN . Then, in this case, without loss of generality we can also assume that Q ⊂ N . Thus Q is also a transversal of N ∩ H in N .
For each r ∈ Im φ, choose, h r ∈ H such that φ(h r ) = r. Then there are r 1 > r 2 > r 3 > . . . such that we can write Therefore, we obtain that Since α φ (x) ∈ Im β φ , we obtain that for each i ≥ 1, Therefore, for each i ≥ 1 and q ∈ Q, a i,q ∈ E[N ∩ H]. This implies, that α φ (x q ) ∈ Im β φ , and so, x q ∈ D E[H,]φ for every q.
The proof of the case N ⊆ H is left as an exercise.
Let H be a normal subgroup of finite index of G and let S be a subset of Let which induces another exact sequence whereα φ andβ φ are homomorphisms of right E[G]-modules defined in the following way. Fix a right transversal Q of H in G. Thenβ φ is defined on a basic tensor bỹ In order to defineα φ , we write an element a ∈ D E[G] as a = q∈Q a q q, with a q ∈ D E[H] , and defineα φ (a) = q∈Q α φ (a q ) ⊗ q.
Observe that with this new notation we also have . Then we put Proof. Let G/K be the maximal torsion-free abelian quotient of G. Let R be a subring of D E[G] generated by D E[K] and G. Then the ring D E[G] is isomorphic to the classical Ore ring of fractions of R. Thus, there are y ∈ R and 0 = z ∈ R such that x = yz −1 . Since x φ = y φ z −1 φ , it is enough to prove the proposition in the case x ∈ R. Thus, let us assume that x ∈ R.
Let A be a transversal of K in G. Then we can write x = a∈A0 x a a, where A 0 is a finite subset of A, and, for each a ∈ A 0 ,x a ∈ D E [K] . Observe that This clearly implies that x φ is a continuous function in φ.

5.4.
Invertibility over Novikov rings. Let H be a normal subgroup of G of finite index and φ ∈ H 1 (H; R). In this subsection we want to give a sufficient condition for Let G 0 be a subgroup of G containing H and let Q be a transversal of H in G 0 . Observe that We can decompose any By the definition, φ,Q has the following properties.
is a restriction of some character of G 0 , then x φ,Q = x φ , and so, in this case φ,Q is multiplicative. However, if φ is an arbitrary character of H 1 (H; R), then φ,Q is not multiplicative in general. This motivates the notion of the defect of φ,Q .
are continuous.
We will use the following properties of φ,Q .
Proof. If g ∈ G 0 , g ∈ Q be such that Hg = Hg. We write z = q∈Q z q q and w = q∈Q w q q, with z q , w q ∈ D E[H] . Then denotes the set of all x ∈ D E[G] such that for some k ≥ 0, U Hn (x) is (G, {H i })-rich for every n ≥ k.
In this section we prove the following theorem. This is the main result of Appendix.
First let us see that K E[G] is a subring of D E [G] . Indeed, if a, b ∈ K E[G] , using Lemma 5.1, we obtain that there exists k ≥ 0 such that for every n ≥ k there is a Thus, in order to show that , we have to prove that for any . First let us consider the case where x ∈ E[G]. Proof. Write x = g∈G α g g and denote by supp x = {g ∈ G : α g = 0}. We will show that x −1 ∈ K E[G] by induction on | supp x|. The base of induction is clear. Let us assume that | supp x| > 1. There exists k ≥ 0 such that |{gH k : g ∈ supp x}| = 1 and |{gH k+1 : g ∈ supp x}| ≥ 2.
Let A be a transversal of H k+1 in H k . Hence, there exists g ∈ G such that we can write x = a∈A x a ag, with x a ∈ E[H k+1 ].
Since g, g −1 ∈ K E[G] , without loss of generality we may assume that g = 1. In particular, x ∈ E[H k ].
For each i ≥ k we fix a transversal Q i of H i in H k . For any a ∈ A we put Claim 5.12. For each i ≥ k, the set V i is rich in H 1 (H i , R).
Proof. First observe that Corollary 5.6 implies that V i,a , and so, V i are open in H 1 (H i , R). Let φ be an irrational character of H 1 (H k , R). Since {H i } is a witnessing chain and φ is irrational, ker φ ≤ H k+1 . Therefore, there exists a ∈ A such that x − x a a φ,Qi = x − x a a φ < (x a a) φ = 1 (x a a) −1 φ = 1 (x a a) −1 φ,Qi .
Since def Qi (φ) = 1, we obtain that φ ∈ V i,a for all i ≥ k, and so V i contains all irrational characters of H k . Now the claim follows from Lemma 5.2.
By the inductive assumption, x a a is invertible in K E [G] . Thus, there exists n ≥ k such that for every i ≥ n and a ∈ A, U Hi ((x a a) −1 ) is rich in H 1 (H i , R). we put By Lemma 5.1, W i is rich. Let φ ∈ W i . Observe that W i is H k -invariant. Hence φ Qi ⊆ V i ∩ a∈A U Hi ((x a a) −1 ). There exists a ∈ A such that φ ∈ V i,a . Observe that x − x a a, x a a, (x a a) −1 ∈ D E[H k ],φ Q i . By Corollary 5.8, x −1 ∈ D E[H k ],φ ⊂ D E[G],φ . Thus, W i ⊆ U Hi (x −1 ) and we are done. Now, we consider the general case.
Proof of Theorem 5.10. We will show that x −1 ∈ K E[G] for every 0 = x ∈ K E[G] by induction on the level l(x) of x, that is defined as follows.
l(x) = min{n − k : x ∈ D E[H k ] and U Hi (x) is rich for every i ≥ n}.
Consider first the case l(x) ≤ 0. Then x ∈ D E[H k ] and U Hi (x) is rich for every i ≥ k. Let H k /K be the maximal torsion-free abelian quotient of H k . Let R be the subring of D E[H k ] generated by D E[K] and H k . Since D E[H k ] is the classical ring of quotients of R, we can write x = yz −1 with non-zero y, z ∈ R. Let A be a transversal of K in H k . Then there are finite subsets A 0 and B 0 of A such that y = a∈A0 y a a, z = a∈B0 z a a with non-zero y a , z a ∈ D E[K] .
Let φ be an irrational character of H k . Observe that takes different values on the elements of A 0 and on the elements of B 0 . Therefore, there are unique a φ ∈ A 0 and b φ ∈ B 0 such that φ(a φ ) = min{φ(a) : a ∈ A 0 } and φ(b φ ) = min{φ(b) : b ∈ B 0 }. Since T −1 ⊆ K E[G] (Proposition 5.11), there exists n such that U Hi (w −1 ) is rich for every w ∈ T and i ≥ n.
For each i ≥ n let Q i be a transversal of H i in H k . For each w ∈ T and i ≥ n we put Observe that V i are open and if φ ∈ H 1 (H k , R), def Qi (φ) = 1. Thus, by Claim 5.13, for all i ≥ n, V i contains all the irrational characters of (U H k (x)) o . Since (U H k (x)) o is rich, Lemma 5.2 implies that V i is rich for i ≥ n.
For each i ≥ n we define . Now, we assume that l(x) > 0 and that the non-zero elements of K E[G] of level less than of l(x) are invertible in K E [G] . There are n and k such that l(x) = n − k, x ∈ D E[H k ] and U Hi (x) is rich for every i ≥ n.
Let A be a transversal of H k+1 in H k . Hence, we can write x = a∈A x a ag, with x a ∈ D E[H k+1 ] .
By Lemma 5.9, for evey a ∈ A, x a ∈ K E[G] and l(x a ) < l(x). For each i ≥ k we fix a transversal Q i of H i in H k . For any a ∈ A we put V i,a = {φ ∈ H 1 (H i , R) : x − x a a φ,Qi · (x a a) −1 φ,Qi < def Qi (φ) −2 }.
Arguing as in the proof of Claim 5.12, we obtain that all V i are rich. By the inductive assumption, x a a is invertible in K E [G] . Thus, there exists n ≥ k such that for every i ≥ n and a ∈ A, U Hi ((x a a) −1 ) is rich in H 1 (H i , R). We put By Lemma 5.1, W i is rich. Let φ ∈ W i . Observe that W i is H k -invariant. Hence φ Qi ⊆ V i ∩ a∈A U Hi ((x a a) −1 ). There exists a ∈ A such that φ ∈ V i,a . Observe that x − x a a, x a a, (x a a) −1 ∈ D E[H k ],φ Q i . By Corollary 5.8, x −1 ∈ D E[H k ],φ ⊆ D E[G],φ . Thus, W i ⊆ U Hi (x −1 ) and we are done.