Subrepresentations in the homology of finite covers of graphs

Abstract Let 
$p \;:\; Y \to X$
 be a finite, regular cover of finite graphs with associated deck group 
$G$
 , and consider the first homology 
$H_1(Y;\;{\mathbb{C}})$
 of the cover as a 
$G$
 -representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group 
$G$
 on the one hand and topological properties of homology classes in 
$H_1(Y;\;{\mathbb{C}})$
 on the other hand. We do so by studying certain subrepresentations in the 
$G$
 -representation 
$H_1(Y;\;{\mathbb{C}})$
 . The homology class of a lift of a primitive element in 
$\pi _1(X)$
 spans an induced subrepresentation in 
$H_1(Y;\;{\mathbb{C}})$
 , and we show that this property is never sufficient to characterize such homology classes if 
$G$
 is Abelian. We study 
$H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$
 —the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in 
$\pi _1(X)$
 . Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with 
$H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$
 .


Introduction
We study subrepresentations in the homology of finite covers of finite graphs, viewed as a representation of the deck group.The subrepresentations we consider arise from homology classes of lifts of closed curves that have interesting topological properties, for example being primitive.One goal is to better understand the connection between topological properties of curves, and representation-theoretic properties of their homology classes.For a finite, regular cover p : Y → X of finite graphs, with associated deck group G, we know that H 1 (Y ; C) is spanned by a finite set of closed loops.Let n be the rank of the free group π 1 (X).For a subset S ⊆ F n = π 1 (X), we consider the subrepresentation , where s, called the preferred elevation of s, is the lift of the smallest power of s such that s is a closed loop in Y , and [s] its homology class.If S is the set of primitive elements in F n , i.e. loops in X that are part of a basis of π 1 (X), we write H prim when n ≥ 3; see [FH16, Propositions 3.2 and 3.3].Malestein and Putman gave a complete answer to this question in [MP19, Theorem C, Example 1.3].For every n ≥ 2 they constructed a finite group which answers the question in the negative.
A central role is played by the theorem of Gaschütz, see [CWH34] or [GLLM15, Theorem 2.1], which identifies the first homology of Y as a representation of G; namely, we have an isomorphism of G-representations where we denote by C triv the trivial one-dimensional representation of G and by C[G] the regular representation of G.
To obstruct primitive homology from being all of homology, Farb and Hensel used the theorem of Gaschütz together with the following representation-theoretic property: If l is a primitive loop in X, then the G-orbit of the homology class of its preferred elevation is linearly independent, in other words Span C[G] {[ l]} ∼ = Ind G φ(l) (C triv ), where φ : π 1 (X) → G is the epimorphism associated to the cover p : Y → X, compare [FH16, Proposition 2.1].From this we obtain a necessary representation-theoretic condition for a homology class to be the homology class of a lift of a primitive element.Recall that a homology class in H 1 (X; Z) can be represented by a primitive loop in X if and only if it is indivisible.The question arises whether there is a similar characterization for homology classes of lifts of primitive elements.Our first result shows that the necessary representation-theoretic property of Farb and Hensel is never sufficient for regular, Abelian covers.
Theorem 1.1.Let G be the deck group of a finite, regular cover Y → X of finite graphs with rank of π 1 (X) at least two.If G is non-trivial Abelian, then there exists z ∈ H 1 (Y ; Z) indivisible such that (i) Span C[G] {z} ∼ = Ind G g (C triv ) for some g ∈ G, and (ii) z is not represented by an elevation of a primitive loop in X.
We ask whether Theorem 1.1 also holds for non-Abelian groups.We extend some of the above questions to other subsets S ⊆ F n .We consider the subset of primitive commutators, by which we mean a commutator of the form [w, w ′ ] for w, w ′ elements in a common basis of F n .Homology classes of elevations of primitive commutators satisfy the following representation-theoretic property, in analogy to [FH16, Proposition 2.1] for primitive elements.
Proposition 1.2.Let G be the deck group of a finite, regular cover Y → X of finite graphs with associated epimorphism φ : π 1 (X) → G. Let x 1 , x 2 ∈ F n be two primitive elements that extend to a free basis of F n .Set x := [x 1 , x 2 ], and let K := φ(x) ≤ φ(x 1 ), φ(x 2 ) =: H ≤ G be the subgroups of G generated by φ(x), respectively φ(x 1 ) and φ(x 2 ).Then we have the following isomorphism of G-representations . Indeed, by transitivity of induction it suffices to realize that C triv is an H-subrepresentation of Ind H K (C triv ) using for example Frobenius reciprocity, compare [Ser77, Chapters 7.1-7.2].
Our next result shows that this representation-theoretic property is not sufficient in the case n = 2. Proposition 1.3.For n = 2 there exists a finite, regular cover p : Y → X and z ∈ H 1 (Y ; Z) ∩ ker(p * ) indivisible, where p * is the induced map on the associated homology groups, such that where G is the deck group associated to the cover p, and (ii) z cannot be represented by an elevation of a primitive commutator in X.
We turn to the space H S 1 (Y ; C) for S the subset of all primitive commutators over all bases of F n (not just some fixed one), and write To obstruct primitive commutator homology to constitute all of ker(p * ), we introduce the following notation following [FH16, Definition 1.3].Let Irr(G) be the set of representatives of the pairwise non-isomorphic irreducible G-representations, and for S ⊆ F n , let Irr S (φ, G) ⊆ Irr(G) be the subset of those irreducible representations V of G that have the property that there is an element in S whose image has a non-zero fixed point.More precisely, V ∈ Irr S (φ, G) if and only if there exists an element s ∈ S and 0 = v ∈ V such that φ(s)(v) = v.If S is the set of primitive elements or primitive commutators, we write Irr prim (φ, G) or Irr comm (φ, G), respectively.For a G-representation M and V ∈ Irr(G), we write M (V ) for the sum of all subrepresentations of M isomorphic to V , and call them the homogeneous components of M .
Theorem 1.4.Let G be the deck group of a finite, regular cover p : Y → X of finite graphs with associated epimorphism φ : π 1 (X) → G. Then The argument follows closely the one of [FH16,Theorem 1.4].The main ingredient of the proof of this theorem is Proposition 1.2.The theorem implies that whenever a group G together with an epimorphism φ : F n → G satisfies Irr comm (φ, G) = Irr(G), we conclude by the theorem of Gaschütz that H comm 1 (Y ; C) = ker(p * ).We will give an example of a group with this property in the case n = 2.In fact, even more is true.
Theorem 1.5.For every n ≥ 2 there exists a finite, regular cover p : Z → X such that H comm Organization.The paper is organized as follows.Section 2 is solely concerned with homology classes of elevations of primitive elements.We will introduce the important results, and finish by giving a proof of Theorem 1.1.In Section 3, we will prove a necessary representation-theoretic property of homology classes of elevations of primitive commutators (Proposition 1.2), but which is not sufficient to characterize those homology classes, see Proposition 1.3.In Section 4, we study the primitive commutator homology, and we will prove Theorem 1.4.We will discuss how primitive homology and primitive commutator homology relate, which will be essential in the proof of Theorem 1.5.
Acknowledgements.I would like to thank Sebastian Hensel for introducing me to this topic and for his helpful remarks.Thanks goes as well to Bram Petri for valuable input, and Peter Feller, Gerhard Hiss and Anna Ribelles Pérez for constructive discussions and feedback.I am grateful to the referee for the detailed comments and suggestions, particularly for pointing out an error in the original argument for Corollary 2.5.

Homology Classes of Elevations of Primitive Elements
Let n ∈ N, n ≥ 2, a finite group G and a surjective homomorphism φ : F n → G, where F n = F x 1 , . . ., x n is the free group on n generators.Let X be the wedge of n copies of S 1 with vertex x 0 .Then we can associate to φ a finite regular path-connected cover p : Y → X with base point y 0 ∈ p −1 (x 0 ) and p * (π 1 (Y, y 0 )) = ker(φ).Under these assumptions, we have G ∼ = F n / ker(φ) ∼ = π 1 (X, x 0 )/p * (π 1 (Y, y 0 )) and G acts on Y by graph automorphisms.This action extends to a linear action of G on the finite-dimensional C-vector space H 1 (Y ; C).
Recall that we call an element of a free (Abelian) group primitive if it is part of a free basis.An element of a free Abelian group is primitive if and only if it is indivisible, that means it cannot be written as a nontrivial multiple of some other element.Being indivisible is equivalent to the coefficient vector with respect to a basis having greatest common divisor one.A primitive element in F n has a primitive image under the quotient map to Z n .On the other hand, every primitive element in Z n has a primitive preimage in F n .
Definition 2.1.For an element l ∈ F n , let k(l) be the minimal number such that l k(l) ∈ ker(φ).The preferred elevation l of l is the lift of l k(l) to Y at the base point y 0 .Lifts of l k(l) at other preimages of x 0 are just called elevations.Unless clear from context we write lp , where p : Y → X is the covering map, to specify which cover we are lifting to.
Note that k(l) is finite for all l ∈ F n , since G is finite.By the regularity of the cover, we obtain all elevations of l by applying the elements of G to the preferred elevation l.We can view l as an element in π 1 (Y, y 0 ), where it will also be called l.Its homology class is denoted by square brackets [ l].It is straightforward to verify that elevations of primitive elements are primitive.
For a cover p : Y → X we denote by p * the induced morphisms on the level of fundamental groups and homology groups.Since Y is finite we have the transfer map p x lift of x x.The map p # commutes with the differential operators and thus induces a map on homology, which will also be denoted by p # .We have p * We summarize the general setup of this article in the following diagram, which can be consulted at every point.

Ab
In the rest of this section we will show that the property given in [FH16, Proposition 2.1] is not sufficient to characterize elevations of primitive elements.We recall the statement here.
Proposition 2.2 (Farb-Hensel).Let G be the deck group of a finite, regular cover p : Y → X of finite graphs with associated epimorphism φ : π 1 (X) → G. Let l be a primitive loop in X and let l be its preferred elevation in Y .We set z := [ l] and g := φ(l).Then there is an isomorphism of G-representations Note that this proposition contains two statements.Firstly, it says that the G-orbit of an elevation of a primitive element is C-linearly independent.Secondly, it tells us that the subgroup over which we induce is related to the primitive element l we started with.It is cyclic and generated by φ(l).If G is Abelian, this allows us to relate z and φ(l), see Equation (1), -a fact which will be exploited in the proof of Theorem 1.1 to verify the insufficiency of this property.
Let V 1 = C triv , . . ., V k(G) be representatives of the isomorphism classes of the irreducible G-representations, where k(G) is the number of nonisomorphic irreducible G-representations.If M is a G-representation, we write M (V i ) for the sum of all subrepresentations of M isomorphic to V i , and call them the homogeneous components of M .Recall that if The following lemma follows directly from the theorem of Gaschütz.
Lemma 2.3.Let M = H 1 (Y ; C).In the above setting, the following properties hold: Proof.(a) The group G acts trivially on p # (H 1 (X; C)), since an element of the deck group permutes lifts.Hence . Since p # is injective as a map on homology we obtain equality by the theorem of Gaschütz and dimensionality reasons.(b) This follows from the theorem of Gaschütz and the fact that the map as C[G]-modules, where G acts trivially on H 1 (X; C), so ker(p * ) does not have simple submodules isomorphic to V 1 .For dimensionality reasons we obtain ker(p Before we prove our first result we state the following lemma about induced representations.
Lemma 2.4.Let G be a finite group and M a representation of G over C. Assume there exists 0 = m ∈ M such that Gm is linearly independent.Let . ., g [G:H] for the left cosets of H in G.For all g ∈ G, g can be uniquely written as g = g i h for some 1 ≤ i ≤ [G : H] and h ∈ H.We then have gm , and C-linear extension.This is an isomorphism of C[G]-modules.
We are now ready to prove our first result.
Proof of Theorem 1.1.Recall that we would like to prove that if G is the deck group of a finite, regular cover Y → X of finite graphs with rank of π 1 (X) at least two, and G is non-trivial Abelian, then there exists an indivisible homology class ) for some g ∈ G, and (ii) z is not represented by an elevation of a primitive loop in X.The proof has two parts: First, we construct an indivisible homology class z ∈ H 1 (Y ; Z) starting from two homology classes of elevations of primitive elements.Second, we verify that z indeed satisfies conditions (i) and (ii).
We will now prove the claim.Since G is Abelian, the map φ factors through Z n , so we obtain a surjective homomorphism φ : Z n → G of Abelian groups.By the invariant factor decomposition of finitely generated modules (refer e.g. to [Bos09, Chapter 2.9, Theorem 2]) the statement is clear for φ because G is non-trivial and n ≥ 2. Since the Abelianization F n → Z n has the property that every primitive element in Z n has a primitive preimage, the claim is proven.
Choose l ′ , l ′′ primitive loops in X that satisfy the above claim.To simplify notation we set g ′ := φ(l ′ ) and g ′′ := φ(l ′′ ) ∈ G.We consider the homology classes z ′ = [ l′ ] and z ′′ = [ l′′ ] ∈ H 1 (Y ; Z) of the preferred elevations l′ and l′′ of the two primitive loops l ′ and l ′′ , respectively.The elements z ′ and z ′′ will be used to build our desired homology class z.Write Let q ∈ Q such that z := q ẑ ∈ H 1 (Y ; Z) is indivisible.This will be our candidate homology class.We come to the second part of the proof.First, we verify that (i) holds for z.For this we need to show that the projection of ẑ to M (V 1 ), namely z ′′ 1 , is non-trivial.Assuming this, it is a straight-forward computation that Stab G (ẑ) = Stab G (z ′ ) = g ′ , and that the G-orbit of ẑ is linearly independent over C, since this is true for z ′ .Now (i) follows from Lemma 2.4 with To show that z ′′ 1 = 0 we will argue by contradiction.In fact, if where the last equality holds since z ′′ is the homology class of the preferred elevation of l ′′ .Thus [l ′′ ] = 0 which contradicts the fact that l ′′ is primitive.
Knowing that b = 0, assume now towards a contradiction that z is the homology class of an elevation of a primitive element l in X.By Proposition 2.2 we have Span C[G] {z} ∼ = Ind G φ(l) (C triv ).On the other hand, by Equation (2), we have Ind G g (C triv ) ∼ = Ind G g ′ (C triv ), with g := φ(l).Now G Abelian implies that g = g ′ , since representations of finite groups over C are determined by their characters and there is an explicit formula for the character of an induced representation, compare e.g.[Isa94, Chapter 5].Further, we compute Again by the indivisibility of the integral homology classes [l] and [l ′′ ], we conclude that they are equal.Since G is Abelian, we obtain g = g ′′ again by Equation (1).This contradicts the fact that g ′ ∩ g ′′ = {1} and g ′ = {1}.
In fact, a slightly stronger result can be verified using the same techniques as in the proof of the above proposition.
Corollary 2.5.Let z ∈ H 1 (Y ; Z) be the element constructed in the proof of Theorem 1.1.Then there is no element in Span C[G] {z} which is the homology class of an elevation of a primitive loop in X.

Proof. Assume there exists z
since G acts as homeomorphisms of the covering space which preserve the projection map p. Then A ∈ Q since both p * (z * ), p * (z) are in H 1 (X; Z).Putting everything together we obtain Since both [l * ] and [l ′′ ] are integral indivisible homology classes, they are equal, and thus g * = g ′′ again by Equation (1).Recall that Stab G

Homology Classes of Elevations of Primitive Commutators
We begin this section by giving a proof of Proposition 1.2, which gives a necessary representation-theoretic property of homology classes of elevations of primitive commutators.
Proof of Proposition 1.2.Recall that we would like to prove that if x 1 and x 2 extend to a free basis of F n and Choose a base point preserving homotopy equivalence from X to the wedge of n circles that maps x 1 and x 2 to two of the circles (e.g. by identifying a maximal spanning tree to a point and then applying a base point preserving homotopy equivalence of the wedge of circles).Then there exists a unique (up to homeomorphism) finite regular cover of the wedge of n circles preserving base points such that the homotopy equivalence lifts to a homotopy equivalence of the corresponding covers and preserves base points.Since homotopy equivalences induce isomorphisms on the level of fundamental and homology groups, we can assume without loss of generality that X is a wedge of n circles and x 1 , x 2 are two of those circles.
The proof now consists of two parts.Firstly we reduce to the case n = 2, where the statement simplifies to Secondly we use surface topology to prove the claim in this special case.We start by reducing to the case n = 2.
Let X 1 be the union of the two circles x 1 and x 2 .Define Y 1 as the component of p −1 (X 1 ) that contains the base point y 0 .Now choose representatives for the left cosets of H in G, say g 1 = 1, . . ., g [G:H] , and set Y i := g i Y 1 for 1 ≤ i ≤ [G : H].These are pairwise disjoint subgraphs of Y , since the vertices of Y correspond to the group elements and every group element is contained in exactly one Y i .It is easy to see that I : , where I is induced by the inclusions of the subgraphs into Y ; see e.g.[FH17,Claim 2.4].Note that I is a morphism of G-representations, since the action of G on the direct sum is given by permuting the summands according to the permutation of the subgraphs {v}, and, since I is injective, we conclude that this sum is in fact direct.Hence . Proposition 1.2 in the case n = 2 applies to H since it is generated by two elements, and we obtain that Span C[H] {v} ∼ = Ind H K (C triv )/C triv .By the exactness of induction (see e.g.[Ser77, Chapter 7.1]), it then follows that Thus, without loss of generality, we can assume that n = 2 and G = H.To prove the proposition in this case, we use surface topology.We would like to show that Span C[G] {[x 12 ]} ∼ = Ind G K (C triv )/C triv .We identify F 2 with the fundamental group of the torus with one boundary component T and base point t 0 ∈ ∂T , with generators x 1 and x 2 represented by the simple closed curves as in the image below.
is represented by α = ∂T .We have a group homomorphism φ : π 1 (T ) ∼ = F 2 → G, and we can associate to ker(φ) a finite, regular, path-connected cover q : S → T with base point s 0 ∈ q −1 (t 0 ) and q * (π 1 (S, s 0 )) = ker(φ).Note that S is again an orientable surface with where the map ψ is given by z → (z, . . ., z), and ϕ sends (z 1 , . . ., z m ) to m i=1 z i [α i ].One can verify that ψ and ϕ are G-equivariant.The exactness follows for example from the exactness of the long exact sequence of the pair (S, ∂S).
Recall that we are interested in understanding Span H 1 (S;C) {[α 1 ], . . ., [α m ]}.Using the short exact sequence it is enough to understand m i=1 C[α i ] and C[∂S] as G-representations.Let us assume without loss of generality that α 1 contains the base point s 0 ∈ S. Then since α is simple.The curves α 1 , . . ., α m are the elevations of α, and G permutes these.Hence we can identify the curves α 2 , . . ., α m with the left cosets of K in G, because K is the stabilizer of α 1 .Choose representatives as G-representations by the defining property of the induced representation.
Clearly, G acts trivially on [∂S].Thus we have proved the proposition, since by the exactness of the short exact sequence above we have It turns out that the above representation-theoretic property is in general not sufficient to characterize homology classes of elevations of primitive commutators.A counter-example for n = 2 will be given in Proposition 1.3, whose construction is based on the idea of stacking covers and the proof of Theorem 1.1.We need some preliminary considerations.
For a group G, a subgroup H ≤ G is characteristic if it is invariant under every automorphism of G, i.e. for all ϕ ∈ Aut(G) we have ϕ(H) ⊆ H.If we have a sequence of subgroups of the form K char H G with K characteristic in H and H normal in G, then K is also normal in G.
Example 3.1.For all n, m ≥ 2 the mod m-homology cover is a characteristic cover of the wedge of n circles.Recall that the mod m-homology cover is given by the surjective group homomorphism It suffices to verify that ker(φ) is characteristic in Since ker(φ) = [x, y], z m | x, y, z ∈ F n , we conclude that α(ker(φ)) ⊆ ker(φ), and hence that the cover is characteristic.
The mod 2-homology cover has the additional property that all primitive commutators lift to primitive elements.Lemma 3.2.Let φ : F 2 → (Z/2Z) 2 be the mod 2-homology cover as in Example 3.1.Then for x := [l, l ′ ] any primitive commutator in F 2 with l = l ′ , its preferred elevation is primitive.
Proof.We denote by p : (Y, y 0 ) → (X, x 0 ) the associated finite, characteristic cover, where (X, x 0 ) is the wedge of two circles and x 0 the point of identification.Let x 1 and x 2 be generators of the fundamental groups of the two circles.We will show the claim for x := [x 1 , x 2 ] first.Then φ(x) = 0 and thus x lifts to a closed curve on Y illustrated by the dotted lines.
We need to extend x to a free basis of π 1 (Y, y 0 ).For example 1 } is a free basis of π 1 (Y, y 0 ).Namely, consider the spanning tree illustrated by the bold lines in the picture above.
For x ′ any primitive commutator we use that there exists a homotopy equivalence of X that sends x ′ to x, since the primitive elements defining x ′ form a basis of F 2 .Since the cover is characteristic, this homotopy equivalence can be lifted to Y , and the same argument works.
There is one last remark before we prove Proposition 1.3.If p : Z → X and q : Y → Z are finite, regular covers with base points z 0 ∈ p −1 (x 0 ) and y 0 ∈ q −1 (z 0 ), respectively, such that p • q : Y → X is a finite, regular cover, then for l any loop on X, we have by the uniqueness of lifts and the choice of base points that lp•q = ( lp ) q .
Proof of Proposition 1.3.Recall that for n = 2 we would like to prove the existence of a finite, regular cover p : Y → X, together with where G is the deck group associated to the cover p, and (ii) z cannot be represented by an elevation of a primitive commutator in X.We will construct Y using iterated covers.Let π 1 (X) ∼ = F 2 be the free group on the generators x 1 and x 2 .Consider the primitive commutators We start with the mod 2-homology cover p 1 : Z → X defined by the surjective homomorphism Then l ′ := x′ and l ′′ := x′′ are primitive in π 1 (Z) by Lemma 3.2.We additionally have that l ′ and l ′′ lie in one basis.For example, the elements x 2 1 , x 2 2 and x 2 x 2 1 x −1 2 complete l ′ and l ′′ to a basis of π 1 (Z) ∼ = F 5 , using the same notation as in the proof of Lemma 3.2.
Consider now the mod 2-homology cover p 2 : Y → Z defined by the surjection and extension.Then p := p 1 •p 2 : Y → X is a finite, regular cover because p 2 is characteristic, and we denote its deck group by G.We have the following short exact sequence of groups 1 We write By the same argument as in the proof of Theorem 1.1, ẑ ∈ H 1 (Y ; Q) and we can choose q ∈ Q such that z := q ẑ ∈ ker(p * ) ≤ H 1 (Y ; Z) is indivisible.Using that N is Abelian, we can now copy the same idea as in the proof of Theorem 1.1 to obtain that z is not the homology class of an elevation of a Example 4.1.For n = 2 any surjective homomorphism φ : F 2 → S 3 , where S 3 denotes the symmetric group on three elements, satisfies Irr comm (φ, S 3 ) = Irr(S 3 ).In fact, the irreducible two-dimensional representation of S 3 does not lie in Irr comm (φ, S 3 ).It can be realized by S 3 acting on V := C 3 / e 1 + e 2 + e 3 by permutation of the coordinates.Since φ is surjective the image of a primitive commutator is either the cycle (123) or (132).Computing the action of (123) on V , we see that one is not an eigenvalue, thus (123) does not have a fixed vector.Similarly for (132).Thus V / ∈ Irr comm (φ, S 3 ).
A more general answer to this question is presented in the following for every n ≥ 2. To see that there are examples where H comm 1 (Y ; C) = ker(p * ) for certain covers p : Y → X, we use iterated covers to relate the question to the one for primitive homology.Note that in general the composition of two regular covers is not again regular, but every finite, regular cover is itself covered by a finite, characteristic cover.This is true by the following observation: if G is a finitely generated group and H G a normal subgroup of finite index k, then there is a subgroup H * ≤ H which is of finite index in G and characteristic in G. Namely, we can take In order to prove that in general H comm 1 = ker(p * ), we reduce the case of primitive commutator homology to the case of primitive homology by looking at the mod 2-homology cover.The proof of Lemma 3.2 can be generalised to show that for every n the mod 2-homology cover as in Example 3.1 has the property that primitive commutators lift to primitive elements.Then we can apply the result of Malestein-Putman in [MP19, Example 1.3] to conclude.For this we need to understand how primitive homology behaves in iterated covers.To avoid confusion we introduce the following notation: If Y → X is a finite, regular cover of finite graphs, we will from now on write H S 1 (Y → X; C) instead of H S 1 (Y ; C).Lemma 4.2.Let q : Z → Y and p : Y → X be finite, regular covers of finite graphs such that p • q : Z → X is a finite, regular cover.Then In other words, primitive homology gets mapped into primitive homology.
Proof.For l a primitive element in π 1 (X), we have for some k ∈ Z q * lp•q = k lp ∈ H prim 1 (Y → X; C).Proof.This is straightforward using that q * : H 1 (Z; C) → H 1 (Y ; C) is surjective.
The last corollary implies that we can assume without loss of generality that the cover constructed by Malestein and Putman in [MP19, Example 1.3] is characteristic by passing to a finite, characteristic cover, which will satisfy that primitive homology is not all of homology.We bring everything together in the following proof.
Proof of Theorem 1.5.Recall that for every n ≥ 2 we would like to construct a finite, regular cover p : Z → X such that H comm 1 (Z → X; C) = ker(p * ).Let p ′ : Y → X be the mod 2-homology cover with group of deck transformations K = (Z/2Z) n .By Example 3.1, we know that this cover is characteristic.Lemma 3.2 tells us that all primitive commutators in X lift to primitive elements in Y .The space Y is again a connected graph with
which proves the first part of the claim.For the second part of the claim we observethat if v 1 , . . ., v [G: g * ] is a C-basis for Span C[G] {z * } ∼ = Ind G g * (Ctriv ) which gets permuted transitively by G and for which Stab G (v 1 ) = g * , then Stab G (v k ) = g * for all k = 2, . . ., [G : g * ], and hence Stab G (z * ) ≤ g * .By a dimension argument we obtain equality.Thus we obtain that g ′ = Stab G (z) ≤ Stab G (z * ) = g * = g ′′ , which contradicts the choice of l ′ and l ′′ in the proof of Theorem 1.1.
the homology class represented by α i and [∂S] := m i=1 [α i ] ∈ H 1 (S; C) the homology class represented by ∂S.Denote by C[α i ] and C[∂S] their respective C-spans.Let m i=1 C[α i ] be the m-dimensional G-representation, where G acts by permuting the homology classes [α i ].We then have the following short exact sequence of G-representations

1(
Y ; C) = ker(p * ).On the other hand, there are examples in rank two with H comm 1 (Y ; C) = ker(p * ), since not every irreducible representation has the property that the image of a primitive commutator has a non-zero fixed point, in other words Irr comm (φ, G) = Irr(G).