Fixed subgroups and computation of auto-fixed closures in free-abelian times free groups

arXiv:1906.02144v1 [math.GR] 5 Jun 2019 FIXED SUBGROUPS AND COMPUTATION OF AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS MALLIKA ROY AND ENRIC VENTURA Abstract. The classical result by Dyer–Scott about fixed subgroups of finite order automorphisms of Fnbeing free factors of Fnis no longer true in Zm×Fn. Within this more general context, we prove a relaxed version in the spirit of Bestvina–Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of m, n. We also study periodic points of endomorphisms of Zm×Fn, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of Zm×Fn. On the way, we prove the analog of Day’s Theorem for real elements in Zm×Fn, contributing a modest step into the project of doing so for any right angled Artin group (as McCool did with respect to Whitehead’s Theorem in the free context). 1. Introduction The goal of this paper is to investigate the properties of fixed point subgroups of automorphisms of direct products of free-abelian and free groups, Zm×Fn. The lattice of subgroups of these groups is quite different from that of free groups, since Zm×Fnis not Howson (i.e., the intersection of two finitely generated subgroups is not necessarily finite generated) as soon as m⩾1 and n⩾2. This affects seriously to the behaviour of the rank function, forcing many situations to degenerate with respect to what happens in free groups. However, there are still several surviving governing rules; we concentrate on some of them, specially about those concerning subgroups fixed by automorphisms of Zm×Fn. Let Gbe a group. We denote by r(G) the rank of G, i.e., the minimal number of generators for G; also, ˜r(G) = max{r(G)−1,0}denotes the reduced rank of G. We denote by End(G) (resp., Aut(G)) the monoid (resp., group) of endomorphisms (resp., automorphisms) of G, and write them all with the arguments on the left, g7→ gα; so, accordingly, αβ denotes the composition g7→ gα 7→ gαβ. Specifically, we will reserve the letter γfor right conjugations, γx:G→G,g7→ x−1gx. We will denote by Mn×m(Z) the n×m(additive) group of matrices over Z, and by GLm(Z) the linear group over the integers. When thinking a matrix Aas a map, it will always act on the right of horizontal vectors, v7→ vA. Given a set S⊆End(G), we let Fix(S) denote the subgroup of Gconsisting of those g∈Gwhich are fixed by every element of S, Fix(S) = {g∈G|gα =g, ∀α∈S}=∩α∈SFix({α}), called the fixed subgroup of S(read Fix(∅) = G). For simplicity, we write Fix φ= Fix({φ}). 1991 Mathematics Subject Classification. 20E05, 20E36, 20K15. Key words and phrases. free-abelian by free, automorphism, fixed subgroup, periodic subgroup, auto-fixed closure. 1 2 MALLIKA ROY AND ENRIC VENTURA For an endomorphism φ∈End(G), define its periodic subgroup as Per ψ=∪∞ p=1 Fix ψp(note that this is always a subgroup since x∈Fix ψpand y∈Fix ψqimply xy ∈Fix ψpq). Observe that Per ψcontains the lattice of subgroups given by Fix ψp,p∈N, with inclusions among them exactly according to divisibility among the exponents: if r|sthen Fix φr⩽Fix φs; and also, if Fix φr⩽Fix φsand d= gcd(r, s) = αr +βs,α, β ∈Z, then Fix φr= Fix φdand d|s. Any direct product of a free-abelian group, Zm,m⩾0, and a free group, Fn,n⩾0, will be called, for short, a free-abelian times free group, G=Zm×Fn. We will work in Gwith multiplicative notation (as it is a non-abelian group as soon as n⩾2) but want to refer to its subgroup Zm⩽G with the standard additive notation (elements thought as row vectors with addition). To make these compatible, consider the standard presentations Zm=ht1,...,tm|[ti, tj], i, j = 1,...,mi and Fn=hz1,...,zn| i, and the standard normal form for elements from Gwith vectors on the left, namely tα1 1···tαm mw(z1,...,zn), where α1,...,αm∈Zand w∈Fnis a reduced word on the alphabet Z={z1,...,zn}; then, let us abbreviate this in the form tα1 1···tαm mw(z1,...,zn) = t(α1,...,αm)w(z1,...,zn) = taw(z1,...,zn), where a= (α1,...,αm)∈Zmis the row vector made with the integers αi’s, and tis a meaningless symbol serving only as a pillar for holding the vector a= (α1,...,αm) up in the exponent. This way, the operation in Gis given by (tau)(tbv) = tatbuv =ta+buv in multiplicative notation, while the abelian part works additively, as usual, up in the exponent. We denote by πthe natural projection to the free part, π:Zm×Fn։Fn,tau7→ u. According to Delgado–Ventura [8, Def. 1.3], a basis of a finitely generated subgroup H⩽fg G is a set of generators for Hof the form {ta1u1,...,tarur, tb1,...,tbs}, where a1,...,ar∈Zm, {u1,...,ur}is a free-basis of Hπ ⩽Fn, and {b1,...,bs}is an abelian basis of LH=L∩Zm⩽Zm. (Note that, to avoid confusions, we reserve the word basis for G, in contrast with abelian-basis and free-basis for the corresponding concepts in Zmand Fn, respectively.) It was showed in [8] that every such subgroup H⩽fg Gadmits a basis, algorithmically computable from any given set of generators. Furthermore, any subgroup H⩽Zm×Fn,n⩾2, is again free-abelian times free, H≃Zm′×Fn′, for some 0 ⩽m′⩽mand some 0 ⩽n′⩽∞(and hence, it is finitely generated if and only if Hπ ⩽Fnis so). We recall from Delgado–Ventura [8, Props. 5.1, 5.2(iii)] that every automorphism Ψ of the group G=Zm×Fn,n⩾2, is of the form Ψ = Ψφ,Q,P :G→G,tau7→ taQ+uabP(uφ), where φ∈Aut(Fn), Q∈GLm(Z), P∈Mn×m(Z), and uab ∈Znis the abelianization of u∈Fn. Furthermore, the composition and inversion of automorphisms work like this: (1) Ψφ,Q,P Ψφ′,Q′,P ′= Ψφφ′,QQ′,P Q′+AP ′,(Ψφ,Q,P )−1= Ψφ−1,Q−1,−A−1P Q−1, where A∈Mn(Z) is the matrix of the abelianization of φ; see [8, Lem. 5.4]. We shall use lowercase Greek letters for endomorphisms of free groups, φ:Fn7→ Fnand uppercase Greek letters for endomorphisms of free-abelian times free groups, Ψ: Zm×Fn7→ Zm×Fn. In particular, Γtau= Γu= Ψγu,Im,0∈Inn(G) is the right conjugation by tau(or, equivalently, by u). The paper is organized as follows. In Section 2, we collect several folklore facts about GLm(Z) for later use; for completeness, we provide proofs highlighting several technical subtleties coming from the fact that Zis not a field, but just an integral domain. In Section 3, we concentrate on finite order automorphisms of Zm×Fnand show that their fixed subgroups are always finitely generated, with rank globally bounded by a computable constant depending only on the ambient ranks m, n (and not depending on the specific automorphism in use); see Theorem 3.2. In Section 4, we turn FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 3 to study periodic points and we manage to extend to free-abelian times free groups a result known to hold both in free-abelian groups and in free groups: the periodic subgroup of an endomorphism equals the fixed subgroup of a high enough power and, furthermore, this exponent can be taken uniform for all endomorphisms, depending only on the ambient ranks m, n; see Theorem 4.3. In Section 5, we consider the auto-fixed closure of a finitely generated subgroup H(roughly speaking, the set of elements fixed by every automorphism fixing H); we prove that it always equals a finite intersection of fixed subgroups, we compute the candidate automorphisms, we decide whether it is finitely generated or not, and in case it is, we effectively compute a basis for it; see Theorem 5.6. As a consequence, we obtain an algorithm to decide whether a given finitely generated subgroup His auto-fixed or not; see Corollary 5.7. To achieve this goal, we make use of a recent result by M. Day about stabilizers of tuples of conjugacy classes in right angled Artin groups being finitely presented, and we prove the analogous version for tuples of exact elements in Zm×Fn. In fact, we only need finite generation and computability of these stabilizers; however, for completeness, we also prove its finite presentability postponing the analysis of the relations (a bit more technical) to the Appendix 6. 2. Preliminaries on GLm(Z) In this section we collect well known and folklore results about the general linear group over the integers, GLm(Z). This group is very well studied in the literature, but we are interested in highlighting several subtleties coming from the fact that Zis not a field, but just an integral domain. Lemma 2.1. Let Q∈GLm(Z)be a matrix such that Qk=Im. Then, we have the decomposition Zm= ker(Q−Im)⊕ker(Qk−1+···+Q+Im). Proof. Since gcd(xk−1+··· +x+ 1, x −1) = 1, Bezout’s equality gives us two polynomials α(x), β(x)∈Z[x] such that 1 = α(x)(xk−1+··· +x+ 1) + β(x)(x−1). Plugging Q, we obtain the matrix equality Im=α(Q)(Qk−1+··· +Q+Im) + β(Q)(Q−Im). Now, for every vector v∈Zm, we have v=vα(Q)(Qk−1+··· +Q+Im) + vβ(Q)(Q−Im). And, since (Q−Im)(Qk−1+··· +Q+Im) = (Qk−1+··· +Q+Im)(Q−Im) = Qk−Im= 0, the first summand is in ker(Q−Im) and the second one in ker(Qk−1+···+Q+Im); hence, Zm= ker(Q−Im) + ker(Qk−1+···+Q+Im). Now let v∈ker(Q−Im)∩ker(Qk−1+···+Q+Im). This means that v(Q−Im) = 0 and v(Qk−1+···+Q+Im) = 0, which imply v=v(Qk−1+···+Q+Im)α(Q) + v(Q−Im)β(Q) = 0. Thus, Zm= ker(Q−Im)⊕ker(Qk−1+···+Q+Im).  Proposition 2.2. Consider the integral linear group GLm(Z),m⩾1. (i) There exists a computable constant L1=L1(m)such that, for every matrix Q∈GLm(Z) of finite order, ord(Q)⩽L1. (ii) There exists a computable constant L2=L2(m)such that, for every matrix Q∈GLm(Z)of finite order, say k= ord(Q)⩽L1, we have that M= Im(Q−Im)is a finite index subgroup of ker(Qk−1+···+Q+Im)with [ker(Qk−1+···+Q+Im) : M]⩽L2. Proof. (i) is a well known fact about integral matrices; we offer here a self-contained proof mixed with that of (ii). Let Q∈GLm(Z) be a matrix of order k < ∞(i.e., Qk=Imbut Qi6=Imfor i= 1,...,k−1). 4 MALLIKA ROY AND ENRIC VENTURA Since (Q−Im)(Qk−1+···+Q+Im) = Qk−Im= 0, we have M= Im(Q−Im)⩽ker(Qk−1+ ···+Q+Im). But, by Lemma 2.1 and the Rank-Nullity Theorem, r(M) = r(Im(Q−Im)) = m−r(ker(Q−Im)) = r(ker(Qk−1+···+Q+Im)) and so, M⩽fi ker(Qk−1+···+Q+Im). This is the index we have to bound globally in terms of m. Let mQ(x) be the minimal polynomial of Q. Since Qk=Im, we have mQ(x)|xk−1 and so, mQ(x) = (x−α1)···(x−αr), where α1...,αrare pairwise different k-th roots of unity (in particular, all roots of mQ(x) are simple and so Qdiagonalizes over the complex field C). Write di= ord(αi). Since cyclotomic polynomials Φdi(x) are irreducible over Z, we deduce Φdi(x)|mQ(x) and so, ϕ(di) = deg(Φdi(x)) ⩽deg(mQ(x)) ⩽m, where ϕis the Euler ϕ-function. But it is well known that limn→∞ ϕ(n) = ∞; see, for example, Dummit–Foote [9, p. 8] from where we can compute a big enough constant C=C(m) such that d1,...,dr⩽C. Finally, k= ord(Q) = lcm(ord(α1)...,ord(αr)) = lcm(d1,...,dr)⩽d1···dr⩽Cr⩽Cm; this is the constant we are looking for in (i), L1=C(m)m. On the other hand, diagonalyzing Q, we get an invertible complex matrix P∈GLm(C) such that P−1QP =D= diag(α1,s1 . . ., α1,...,αr,sr . . ., αr), where s1,...,srare the multiplicities in the characteristic polynomial, χQ(x) = (x−α1)s1···(x−αr)sr. Since αiis a primitive di-th root of unity, it can take ϕ(di)⩽mmany values and, since s1+···+sr=m, the diagonal matrix Dcan take only finitely many values; we can make a list of all of them (up to reordering of the αi’s) and, for each one, compute the index [ker(Dk−1+···+D+Im) : Im(D−Im)]. The maximum of these indices is the constant L2=L2(m) we are looking for in (ii), because [ker(Qk−1+···+Q+Im) : M] = [(ker(Qk−1+···+Q+Im))P: (Im(Q−Im))P] = = [ker P−1(Qk−1+···+Q+Im)P: Im(P−1(Q−Im)P)] = [ker(Dk−1+···+D+Im) : Im(D−Im)].  We study now the periodic subgroup of a matrix Q∈Mm(Z), namely Per Q={v∈Zm|vQp= v, for some p⩾1}. The next Proposition states that a uniform single exponent depending only on m,L3=L3(m), is enough to capture all the periodicity of all m×mmatrices Q. Proposition 2.3. There exists a computable constant L3=L3(m)such that Per Q= Fix QL3, for every Q∈Mm(Z). Proof. As we argued in the proof of Proposition 2.2(i), there is a computable constant C=C(m) such that ϕ(d)> m for every d > C(m); see Dummit–Foote [9, p. 8]. Let us prove that the statement is true with the constant L3=C(m)! Fix a matrix Q∈Mm(Z), and consider its characteristic polynomial factorized over the complex field C,χQ(x) = (x−α1)s1···(x−αr)sr, where αi6=αj,i6=j. Standard linear algebra tells us that Cm=Kα1⊕ · · · ⊕ Kαr, where Kαi= ker(Q−αiIm)si⩽Cmis the generalized eigenspace of Qwith respect to αi, a Q-invariant C-subspace of Cm. Distinguish now between those αi’s which are roots of unity, say α1,...,αr′, and those which are not, say αr′+1,...,αr, 0 ⩽r′⩽r. Write di= ord(αi), for i= 1,...,r′, and observe that d1,...,dr′⩽C(since the cyclotomic polynomials Φdi(x) are Q-irreducible and so must divide χQ(x)∈Z[X], which has degree m); in particular, αL3 i= 1, i= 1,...,r′. Now, let v∈Per Q, i.e., vQp=vfor some p⩾1. Applying the above decomposition, v= v1+···+vr, where vi∈Kαi, and the Q-invariance of Kαi, we get the alternative decomposition FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 5 v=vQp=v1Qp+···+vrQp. So, viQp=vi, i.e., vi(Qp−Im) = 0, for i= 1,...,r. For a fixed i, distinguish the following two cases: (i) if αp i6= 1, then αiis not a root of xp−1 and so, 1 = gcd (x−αi)si, xp−1. By Bezout’s equality, there are polynomials a(x), b(x)∈C[x] such that 1 = (x−αi)sia(x)+(xp−1)b(x). Plugging the matrix Qand multiplying by the vector vion the left, we obtain vi=vi(Q− αiIm)sia(Q) + vi(Qp−Im)b(Q) = 0. (ii) if αp i= 1, then x−αi= gcd (x−αi)si, xp−1. By Bezout’s equality, there are polynomials a(x), b(x)∈C[x] such that x−αi= (x−αi)sia(x)+(xp−1)b(x). Now, plugging the matrix Qand multiplying by the vector vion the left, we have vi(Q−αiIm) = vi(Q−αiIm)sia(Q)+ vi(Qp−Im)b(Q) = 0. That is, viQ=αiviand so, viQL3=αL3 ivi=vi. Altogether, v=v1+···+vr=Pi|αp i=1 viand vQL3=Pi|αp i=1 viQL3=Pi|αp i=1 viQL3= Pi|αp i=1 vi=v, and v∈Fix QL3. This completes the proof that Per Q= Fix QL3. 3. Finite order automorphisms of Zm×Fn A well-known (and deep) result by Bestvina–Handel [2] establishes a uniform bound (in fact, the best possible) for the rank of the fixed subgroup of any automorphism of Fn: for every φ∈Aut(Fn), r(Fix φ)⩽n. This result followed an interesting previously know particular case due to Dyer– Scott [10]: if φ∈Aut(Fn) is of finite order then Fix φis a free factor of Fn. When we move to a free-abelian times free group, G=Zm×Fn, the situation degenerates, but still preserving some structure. In Delgado–Ventura [8], the authors gave an example of an automorphism Ψ ∈Aut(G) with Fix Ψ not being finitely generated; so, there is no possible version of Bestvina–Handel result in G. Following the parallelism, we show below an example of an automorphism Ψ ∈Aut(G) of finite order (in fact, of order 2) such that Fix Ψ is not a factor of G; see Example 3.3. However, as a positive result, in Theorem 3.2(ii) below we prove that finite order automorphisms of Gdo have finitely generated fixed subgroups, in fact with a computable uniform upper bound for its rank, in terms of mand n. Lemma 3.1. Let G=Zm×Fn. For given finitely generated subgroups H⩽fg K⩽fg G, the following are equivalent: (a) every basis of Hextends to a basis of K; (b) some basis of Hextends to a basis of K; (c) Hπ ⩽ff Kπ and LH⩽⊕LK. In this case, we say that His a factor of K, denoted H⩽fK; this is the notion in Gcorresponding to free factor in Fn(denoted ⩽ff ), and direct summand in Zm(denoted ⩽⊕). Proof. (a)⇒(b) is obvious. Assuming (b), we have H=hta1u1,...,tarur, tb1,...,tbsiand K=hta1u1,...,tarur, tar+1 ur+1, ...,tar+pur+p, tb1,...,tbs, tbs+1 ,...,tbs+qi, where {u1,...,ur}is a free-basis of Hπ,{b1,...,bs}is an abelian-basis of LH,{u1, . . . , ur+p}is a free-basis of Kπ, and {b1,...,bs+q}is an abelian-basis of LK. Therefore, Hπ ⩽ff Kπ and LH⩽⊕LK. This proves (b)⇒(c). Finally, assume (c). Given any basis {ta1u1,...,tarur, tb1,...,tbs}for H,{u1,...,ur}is a free-basis of Hπ (which can be extended to a free-basis {u1,...,ur, ur+1,...,ur+p}of Kπ since 6 MALLIKA ROY AND ENRIC VENTURA Hπ ⩽ff Kπ); and {b1,...,bs}is an abelian-basis of LH(which can be extended to an abelian-basis {b1,...,bs, bs+1,...,bs+q}of LKsince LH⩽⊕LK). Then, choose vectors ar+1,...,ar+p∈Zm such that tar+1 ur+1, . . . , tar+pur+p∈K(this is always possible because ur+1,...,ur+p∈Kπ), and {ta1u1,...,tarur, tar+1 ur+1,...,tar+pur+p, tb1,...,tbs, tbs+1 ,...,tbs+q}is a basis of K(in fact, they generate K, and have the appropriate form). This proves (c)⇒(a).  Theorem 3.2. Let G=Zm×Fn,m, n ⩾0. (i) There exists a computable constant C1=C1(m, n)such that, for every Ψ∈Aut(G)of finite order, ord(Ψ) ⩽C1. (ii) There exists a computable constant C2=C2(m, n)such that, for every Ψ∈Aut(G)of finite order, r(Fix Ψ) ⩽C2. Proof. (i). By Proposition 2.2(i), the set {ord(Q)|Q∈GLm(Z) of finite order}is bounded above by a computable constant L1(m). And by Lyndon–Schupp [13, Cor. I.4.15], {ord(φ)|φ∈ Aut(Fn) of finite order} ⊆ {ord(Q)|Q∈GLn(Z) of finite order}, which is bounded above by L1(n). If n⩽1 then G=Zm+nis free-abelian and the constant C1=L1(m+n) makes the job; if m= 0 then G=Fnis free and the constant C1=L1(n) makes the job. So, suppose m⩾1, n⩾2, and take an automorphism Ψ = Ψφ,Q,P ∈Aut(G). By Delgado– Ventura [8, Lemma 5.4(ii)], Ψk φ,Q,P = Ψφk,Qk,Pk, where Pk=Pk−1 i=0 AiPQk−1−iand A∈GLn(Z) is the abelianization of φ. In particular, if Ψ is of finite order then φand Qare so too; furthermore, ord(Ψ) = λr3, where r3= lcm(r1, r2), r1= ord(φ), and r2= ord(Q). But Ψr3= Ψid,id,Pr3 and Ψλr3= (Ψid,id,Pr3)λ= Ψid,id,λPr3. Hence, Ψ is either of order r3or of infinite order. In other words, {ord(Ψ) |Ψ∈Aut(G) of finite order} ⊆ {lcm(ord(φ),ord(Q)) |φ∈Aut(Fn), Q ∈ GLm(Z),both of finite order}, which is bounded above by the constant C1(m, n) = L1(n)L1(m). (ii). If n⩽1 then C2=m+nmakes the job, if m= 0 then C2=nmakes the job. So, suppose m⩾1, n⩾2. Delgado–Ventura [8, §6] discusses the form of the fixed subgroup of a general automorphism Ψφ,Q,P ∈Aut(G), namely, LFix Ψ = Fix(Q) = E1(Q) (the eigenspace of eigenvalue 1 for Q), and (Fix Ψ)π=NP′−1ρ′−1, where ρ:Fn։Znis the abelianization map, ρ′is its restriction to Fix φ,P′is the restriction of Pto Im ρ′,M= Im(Q−Im), N=M∩Im P′, and (Fix Ψ)π=NP ′−1ρ′−1EFix φ⩽Fn, see the following diagram, (2) ⩾M= Im(Q−Im) =M∩Im P′. ⩽ E E FnZn ρ////Zm P// Fix φIm ρ′ ρ′ ////Im P′ P′ //// E E E Q−Im  N NP′−1✤ oo NP′−1ρ′−1✤ oo (Fix Ψ)π= If Fix φis trivial or cyclic, then r(Fix Ψ) = r((Fix Ψ)π)+r(E1(Q)) ⩽1+m. So, taking C2(m, n)⩾ 1 + m, we are reduced to the case r(Fix φ)⩾2. FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 7 With this assumption, (Fix Ψ)π6= 1 (it always contains the commutator of Fix φ) and so, Fix Ψ ⩽ Gis finitely generated if and only if (Fix Ψ)π⩽Fnis so, which is if and only if the index ℓ:= [Fix φ: (Fix Ψ)π] = [Fix φ:NP′−1ρ′−1] = [Im ρ′:NP′−1] = [Im P′:N] is finite. In this case, by the Schreier index formula, ˜r(Fix Ψ) = ˜r((Fix Ψ)π) + r(E1(Q)) ⩽ℓ˜r(Fix φ) + m⩽ℓ(n−1) + m. Therefore, we are reduced to bound the index ℓin terms of nand m. First, let us prove that Ψ being of finite order implies ℓ= [Im P′:N]<∞. Put k= ord(Ψφ,Q,P ) so, φk= Id, Qk=Im, and Pk=Pk−1 i=0 AiPQk−1−i= 0, where A∈GLn(Z) is the abelianization of φ. By Proposition 2.2(ii), the subgroup M= Im(Q−Im) is a finite index subgroup of ker(Qk−1+···+Q+Im), with the index bounded above by a computable constant depending only on m, [ker(Qk−1+···+Q+Im) : M]⩽L2(m). We claim that Im P′⩽ker(Qk−1+···+Q+Im). In fact, take u∈Fix φ, note that uφ =u and so (uρ′)A=uφρ′=uρ′, and split (uρ′)P′=v1+v2, with v1∈ker(Q−Im) and v2∈ ker(Qk−1+···+Q+Im); see Lemma 2.1. Multiplying by Qk−1+···+Q+Imon the right, v1(Qk−1+···+Q+Im) = (v1+v2)(Qk−1+···+Q+Im) = (uρ′)P′(Qk−1+···+Q+Im) = = k−1 X i=0 (uρ′)PQk−1−i= k−1 X i=0 (uρ′)AiPQk−1−i= (uρ′) k−1 X i=0 AiPQk−1−i= (uρ′)Pk= 0, from which we deduce v1∈ker(Q−Im)∩ker(Qk−1+···+Q+Im) = {0}so, (uρ′)P′=v2∈ ker(Qk−1+···+Q+Im). Therefore, Im P′⩽ker(Qk−1+···+Q+Im). Finally, intersecting the inclusion M⩽fi ker(Qk−1+···+Q+Im) with Im P′, we get N= M∩Im P′⩽fi Im P′, and ℓ= [Im P′:N]⩽[ker(Qk−1+···+Q+Im) : M]⩽L2(m). Hence, taking C2(m, n)⩾L2(m)(n−1) + mwill suffice for the present case. Therefore, C2(m, n) = L2(m)(n−1) + m+ 1 serves as the upper bound claimed in (ii).  Example 3.3. Here is an example of an order 2 automorphism of G=Z2×F3whose fixed subgroup is not a factor of G. Consider the automorphism Ψφ,Q,P determined by φ:F3→F3,z17→ z−1 1, z27→ z2,z37→ z3,Q=1 0 0−1∈GL2(Z), and P=1 0 0 1 0 2 ∈M3×2(Z), i.e., Ψ: Z2×F3−→ Z2×F3 z17−→ t(1,0)z−1 1 z27−→ t(0,1)z2 z37−→ t(0,2)z3 t(1,0) 7−→ t(1,0) t(0,1) 7−→ t(0,−1). An easy computation shows that Ψ2= Id, i.e., Ψ has order 2. To compute Fix Ψ, let us follow diagram (2): first note that Fix φ=hz2, z3i; so, Im ρ′=h(0,1,0),(0,0,1)i, Im P′=h(0,1),(0,2)i= h(0,1)i. On the other hand, M=h(0,2)i,N=h(0,2)i, and NP ′−1=h(0,2,0),(0,0,1)i. Therefore, (Fix Ψ)π=NP ′−1ρ′−1={w(z2, z3)| |w|z2even}=hz2 2, z3, z−1 2z3z2i. So, solving the systems of equations to compute the vectors associated with each element of the free part, we obtain that t(0,1)z2 2, t(0,1)z3, t(0,1)z−1 2z3z2∈Fix Ψ. Finally, since (Fix Ψ) ∩Z2=E1(Q) = h(1,0)i, we deduce that Fix Ψ = ht(0,1)z2 2, t(0,1)z3, t(0,1)z−1 2z3z2, t(1,0)i. Since Hπ =hz2 2, z3, z−1 2z3z2iis not a free factor of F3, Fix Ψ is not a factor of Z2×F3; see Lemma 3.1. 8 MALLIKA ROY AND ENRIC VENTURA Theorem 3.2 has the following easy corollary: Corollary 3.4. Let Ψ∈End(Zm×Fn). If Fix Ψpis finitely generated then Fix Ψ is also finitely generated; the converse is not true. Proof. Clearly, Ψ restricts to an automorphism Ψ|∈Aut(Fix Ψp) such that Fix Ψ|= Fix Ψ and (Ψ|)p= Id. Since Fix Ψpis finitely generated, we have Fix Ψp≃Zm′×Fn′for some m′⩽mand n′<∞and, applying Theorem 3.2(ii), we get r(Fix Ψ) = r(Fix Ψ|)<∞(in fact, bounded above by C2(m′, n′)). The converse is not true as the following example shows. Consider Ψ: Z×F2→Z×F2,z17→ tz−1 1, z27→ z−1 2,t7→ t−1. It is straightforward to see that Fix Ψ = 1. But Ψ2:Z×F2→Z×F2, z17→ t−2z1,z27→ z2,t7→ tand so, Fix Ψ2=hti × {w(z1, z2)∈F2| |w|z1= 0}=hti × hhz2ii is not finitely generated.  4. Periodic points of endomorphisms of Zm×Fn Corollary 3.4 states that, for Ψ ∈Aut(G), the lattice of fixed subgroups of powers of Ψ could simultaneously contain finitely and non-finitely generated subgroups but, as soon as one of them is finitely generated, the smaller ones must be so. In the abelian case G=Zm, this lattice of fixed subgroups is always finite, and coming from a set of exponents uniformly bounded by m; this is precisely the contents of Proposition 2.3. In the free case, combining results from Bestvina–Handel, Culler, Imrich–Turner, and Stallings, the exact analogous statement is true: Proposition 4.1 (Bestvina–Handel–Culler–Imrich–Turner–Stallings [2, 6, 12, 19]; see also [3, Prop. 3.1]).For every φ∈End(Fn), we have Per φ= Fix φ(6n−6)!. Proof. Culler [6] proved that every finite order element in Out(Fn) has order dividing (6n−6)!; and the same is true in Aut(Fn) since the natural map Aut(Fn)։Out(Fn) has torsion-free kernel. On the other hand Stallings [19] proved that, for every φ∈Aut(Fn), there exists s⩾0 such that Per φ= Fix φs. Also, Imrich–Turner [12] proved that the so-called stable image of an endomorphism φ∈End(Fn), namely Fφ∞=∩∞ p=1Fnφp, has rank at most n, it is φ-invariant, it contains Per φ, and the restriction φ|:F φ∞→Fnφ∞is bijective. Finally, Bestvina–Handel Theorem (see [2]) estates that r(Fix φ)⩽n, for any φ∈Aut(Fn). Combining these four results we can easily deduce the statement: given an endomorphism φ:Fn→Fn, consider its restrictions φ1:Fnφ∞→Fnφ∞and φ2: Per φ1→Per φ1, both bijective; furthermore, Per φ2= Per φ1= Fix φs 1(assume s⩾0 minimal possible), r(Per φ1)⩽r(Fφ∞)⩽n, and φ2has order s. Therefore, sdivides (6 r(Per φ1)−6)! and so (6n−6)! as well. We conclude that Per φ= Per φ1= Fix φs 1= Fix φs⩽Fix φ(6n−6)! ⩽Per φand so, Per φ= Fix φ(6n−6)!. Remark 4.2. Modulo missing details, this fact was implicitly contained in an older result by M. Takahasi, who proved that an ascending chain of subgroups of a free group, with rank uniformly bounded above by a fixed constant (like the Fix ψp’s), must stabilize; see [13, p. 114]. We close the present section by extending this same result to the context of free-abelian times free groups. FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 9 Theorem 4.3. There exists a computable constant C3=C3(m, n)such that Per Ψ = Fix ΨC3, for every Ψ∈End(Zm×Fn). Proof. Delgado–Ventura [8, Prop. 5.1] gave a classification of all endomorphisms of G=Zm×Fnin two types. For those of the second type, say Ψz,l,h,Q,P (see [8] for the notation), it is clear that the subgroup hz, Zmi⩽Zm×Fnis invariant under Ψ (denote Ψ|:hz, Zmi → hz, Zmiits restriction), and it contains Im Ψ. Therefore, by Proposition 2.3, Per Ψ = Per Ψ|= Fix(Ψ|)L3(m+1) = Fix ΨL3(m+1), since hz, Zmi ≃ Zm+1 is abelian. Thus, the computable constant C3(n, m) = L3(m+ 1) satisfies the desired result for all endomorphisms of the second type. Suppose now that Ψ is of the first type, i.e., Ψ = Ψφ,Q,P , where φ∈End(Fn), Q∈Mm×m(Z), and P∈Mn×m(Z). By Propositions 2.3 and 4.1, we know that Per Q= Fix QL3and Per φ= Fix φ(6n−6)! for some computable constant L3=L3(m). Take C3(m, n) = lcm L3(m),(6n−6)! and let us prove that PerΨ = Fix ΨC3. By construction, we have both Per Q= Fix QC3and Per φ= Fix φC3. It remains to see that the matrix Pdoes not affect negatively into the calculations. To prove Per Ψ = Fix ΨC3, it is enough to see that Fix Ψk⩽Fix ΨC3for all k⩾1, which reduces to see that Fix ΨλC3⩽Fix ΨC3for every λ∈N(in fact, if this is true then Fix Ψk⩽Fix ΨkC3⩽Fix ΨC3, for an arbitrary k⩾1). By Delgado–Ventura [8, Lemma 5.4(ii)], powers work like this: (Ψφ,Q,P )k= Ψφk,Qk,Pk, where Pk=Pk−1 i=0 AiPQ(k−1)−iand A∈Mn×n(Z) is the abelianization matrix corresponding to φ∈ End(Fn). In our situation, (Ψφ,Q,P )C3= ΨφC3,QC3,PC3, and (Ψφ,Q,P )λC3= ΨφλC3,QλC3,PλC3, where (3) PλC3=PλC3−1 i=0 AiPQ(λC3−1)−i =Pλ−1 j=0 PC3−1 i=0 AjC3+iP Q(λC3−1)−(jC3+i) =Pλ−1 j=0 PC3−1 i=0 AjC3+iP Q(λ−j)C3−1−i =Pλ−1 j=0 AjC3PC3−1 i=0 AiPQ(C3−1)−iQ(λ−j−1)C3 =Pλ−1 j=0 (AC3)jPC3(QC3)(λ−1)−j. Take any element tau∈Fix ΨλC3and let us prove that tau∈Fix ΨC3. Our assumption means that taQλC3+uabPλC3(uφλC3) = tauand so, (1) a(Im−QλC3) = uabPλC3, and (2) u∈Fix φλC3⩽Per φ= Fix φC3; in particular, uabAC3=uab. Now from (3) and condition (1) we have, a(Im−QC3)(I+QC3+···+Q(λ−1)C3) = uab Pλ−1 j=0 (AC3)jPC3(QC3)(λ−1)−j =uab Pλ−1 j=0 PC3(QC3)(λ−1)−j =uabPC3Pλ−1 j=0 (QC3)(λ−1)−j =uabPC3I+QC3+···+Q(λ−1)C3, which means that a(Im−QC3)−uabPC3∈ker Im+QC3+···+Q(λ−1)C3. But ker Im+QC3+···+Q(λ−1)C3⩽ker(Im−QλC3) = Fix QλC3⩽Per Q= Fix QC3= ker(Im−QC3) 16 MALLIKA ROY AND ENRIC VENTURA Now Fix Ψ1∩ · · · ∩ Fix Ψk⩽Gis finitely generated if and only if (Fix Ψ1∩ · · · ∩ Fix Ψk)π= N˜ P′−1ρ′−1is finitely generated, which (since it is a normal subgroup) happens if and only if N˜ P′−1ρ′−1is trivial (i.e., Fix φ1∩ · · · ∩ Fix φk=huiwith uρ 6= 0 and N={0}) or of finite index in Fix φ1∩ · · · ∩ Fix φk. That is, Fix Ψ1∩ · · · ∩ Fix Ψkis finitely generated if and only if (i) Fix φ1∩ · · · ∩ Fix φk=huiwith uρ 6= 0 and N={0}, or (ii) [Im P′:N] = [Im ρ′:N˜ P′−1] = [Fix φ1∩ · · · ∩ Fix φk:N˜ P′−1ρ′−1]<∞or, equivalently, r(N) = r(Im ˜ P′). These conditions can effectively be checked by computing a free-basis for Fix φ1∩ · · · ∩ Fix φkwith Theorem 5.1 and pull-backs of graphs, and then computing the ranks r(Im ˜ P′) and r(N) with basic linear algebra techniques. So, we can effectively decide whether Fix Ψ1∩ · · · ∩ Fix Ψkis finitely generated or not. Finally, let us assume it is so, and let us compute a basis for Fix Ψ1∩ · · · ∩ Fix Ψk. If we are in the situation (i) then Fix φ1∩ · · · ∩ Fix φk=hui,uρ 6= 0, and M∩Im ˜ P′=N={0} so, the only elements in Fix Ψ1∩· · ·∩Fix Ψkare those of the form taurwith a(Im−˜ Q) = r·uρ ˜ P= 0. That is, Fix Ψ1∩ · · · ∩ Fix Ψk=hu, td1,...,tdsiwhere hd1,...,dsi=E1(Q1)∩ · · · ∩ E1(Qk)⩽Zm. If we are in situation (ii), then we can compute a set {c1,...,cq} ⊂ Znof coset representatives of N˜ P′−1in Im ρ′, namely Im ρ′= (N˜ P′−1)c1⊔ · · · ⊔ (N˜ P′−1)cq. Having computed a free-basis {v1,...,vp}for Fix φ1∩ · · · ∩ Fix φk, we can choose arbitrary preimages y1,...,yqof c1,...,cqup in Fix φ1∩ · · · ∩ Fix φk, and we get a set of right coset representatives of (Fix Ψ1∩···∩Fix Ψk)π= N˜ P′−1ρ′−1in Fix φ1∩ · · · ∩ Fix φk, (6) Fix φ1∩ · · · ∩ Fix φk= (N˜ P′−1ρ′−1)y1⊔ · · · ⊔ (N˜ P′−1ρ′−1)yq. Now, we build the Schreier graph for N˜ P′−1ρ′−1⩽fi Fix φ1∩· · ·∩Fix φkwith respect to {v1,...,vp} in the following way: (1) take the cosets from (6) as vertices, and with no edge; (2) for every vertex (N˜ P′−1ρ′−1)yiand every letter vj, add an edge labeled vjfrom (N˜ P′−1ρ′−1)yito (N˜ P′−1ρ′−1)yivj, algorithmically identified among the available vertices by repeatedly solving the membership problem for N˜ P′−1ρ′−1(note that we can easily do this by abelianizing the candidate and checking whether it belongs to N˜ P′−1). Once we have run over all i= 1,...,q and all j= 1,...,p, we have computed the full (and finite!) Schreier graph, from which we can select a maximal tree and obtain a free-basis {u1,...,ur}for the subgroup corresponding to closed paths at the basepoint, i.e., for N˜ P′−1ρ′−1= (Fix Ψ1∩ · · · ∩ Fix Ψk)π. Finally, solving linear systems of equations (which must be mandatorily compatible), we obtain vectors e1,...,er∈Zmsuch that te1u1,...,terur∈Fix Ψ1∩ · · · ∩ Fix Ψk. We conclude that {te1u1,...,terur, td1,...,tds}is a basis for Fix Ψ1∩ · · · ∩ Fix Ψk. Proof of Theorem 5.6. From the given generators, compute a basis for H, say {ta1u1,...,tarur, tb1,...,tbs}. Now, using Theorem 5.12, we can compute automorphisms Ψ1,...,Ψk∈Aut(G) such that AutH(G) = hΨ1, . . . , Ψki. So, we have that a-ClG(H) = Fix Ψ1∩ · · · ∩ Fix Ψk. Finally, using Proposition 5.13, we can decide whether this intersection is finitely generated or not and, in the affirmative case, compute a basis for it.  Proof of Corollary 5.7. Given generators for H⩽fg G, apply Theorem 5.6. If a-ClG(H) is not finitely generated then conclude that His not auto-fixed. Otherwise, we get a set of automorphisms Ψ1,...,Ψk∈Aut(G) such that a-ClG(H) = Fix Ψ1∩ · · · ∩ Fix Ψk, and a basis for a-ClG(H)⩾H. FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 17 Now His auto-fixed if and only if this last inclusion is an equality (which can be algorithmically checked by using a solution to the membership problem in G; see [8, Prop. 1.11]); and in this case, Ψ1,...,Ψkare the automorphisms such that H= Fix Ψ1∩ · · · ∩ Fix Ψk. 6. Appendix: computation of relations Let us go back to the details of the proof of Theorem 5.12 and complete it by computing a finite set of defining relations for AutH(G). Proof of Theorem 5.12 continued (relations part). We have already computed a finite set of generators for AutH(G). To find the defining relations, we distinguish again the cases r= 0, r⩾2, and r= 1 (in increasing order of difficulty): •Case 1: r= 0. Here, we have H=LH, and we know that AutH(G) is (finitely) generated by the automorphisms of Gof the form (1) Ψφ,Im,0, with φrunning over the Nielsen automorphisms of Fn; (2) Ψid,Q,0, with Qrunning over the generators of AutLH(Zm); and (3) Ψid,Im,1i,j , with i= 1,...,n, j= 1,...,m. Therefore, from [8, Thm. 5.5], we deduce that AutH(G)≃Mn×m⋊AutLH(Zm)× Aut(Fn)with the natural action. Hence, we can easily compute an explicit finite presentation for this group by using the presentation for AutLH(Zm) we got from Day’s Theorem 5.10, any know presentation for Aut(Fn) (see, for example, [1]), and the standard presentation for Mn×m≃Znm. •Case 2: r⩾2. In this case, we already know that AutH(G) = hΨ′ 1,...,Ψ′ ℓi. Let us find a complete set of defining relations for this set of generators. Observe first that, for every Ψ ∈AutW(G), the decomposition Ψ = Ψ′Γxmentioned in (4) is unique: if Ψ′Γx= Ψ′′Γy, with Ψ′,Ψ′′ ∈AutH(G) and x, y ∈Fn, then x−1u1x=y−1u1yand x−1u2x=y−1u2y, which implies that xy−1commutes with the freely independent elements u1, u2 and so, xy−1= 1; hence, Γx= Γyand Ψ′= Ψ′′. In other words, AutH(G)∩Inn(G) = {IdG}and so, AutW(G)/Inn(G) = AutH(G) Inn(G)/Inn(G)≃AutH(G)/AutH(G)∩Inn(G)= AutH(G). We have the following two sources of natural relations among the Ψ′ i’s. From (5), for each i= 1,...,d we have IdG=Ri(Ψ1,...,Ψℓ) = Ri(Ψ′ 1Γx1,...,Ψ′ ℓΓxℓ) = Ri(Ψ′ 1,...,Ψ′ ℓ)Γyi= Ri(Ψ′ 1,...,Ψ′ ℓ), where yi∈Fnmust be 1, again, because r⩾2. On the other hand, for each one of the ngenerating letters of Fn, say z1,...,zn, compute an expression for the conjugation Γzj∈Inn(G)⩽AutW(G) in terms of Ψ1,...,Ψℓ, say Γzj=Sj(Ψ1,...,Ψℓ), and we have Γzj=Sj(Ψ1,...,Ψℓ) = Sj(Ψ′ 1Γx1,...,Ψ′ ℓΓxℓ) = Sj(Ψ′ 1,...,Ψ′ ℓ)Γyjfor some yj∈Fn; but then IdG=Sj(Ψ′ 1,...,Ψ′ ℓ)Γyjz−1 j=Sj(Ψ′ 1,...,Ψ′ ℓ), j= 1,...,n, gives us a second set of relations for AutH(G) (here, again, yjz−1 j= 1 since r⩾2). Therefore, AutH(G) = AutW(G)/Inn(G) =hΨ1,...,Ψℓ|R1,...,Rdi/Inn(G) =hΨ′ 1,...,Ψ′ ℓ|R1,...,Rd, S1,...,Sni. (Note that w(Ψ1,...,Ψℓ)7→ w(Ψ′ 1,...,Ψ′ ℓ) or, equivalently, Ψ 7→ Ψ′= ΨΓx−1for the unique possible x∈Fn, is the canonical projection AutW(G)։AutH(G)≃AutW(G)/Inn(G).) •Case 3: r= 1. Here, H=htau, tb1,...,tbsi⩽Gwith 1 6=u∈Fn(for notational simplicity, we have deleted the subindex 1 from uand a). This case is a bit more complicated than Case 2 18 MALLIKA ROY AND ENRIC VENTURA because the decomposition Ψ = Ψ′Γxfrom (4) is not unique now; additionally, AutH(G) contains some non-trivial conjugation, namely Γˆu, and so we cannot mod out Inn(G) from AutW(G) because this would kill part of AutH(G). In the present case, we know that AutH(G) = hΨ′ 1,...,Ψ′ ℓ,Γˆui. Let us adapt the two previous sources of natural relations among them, and discover a third one. From (5), for each i= 1,...,d we have IdG=Ri(Ψ1,...,Ψℓ) = Ri(Ψ′ 1Γx1,...,Ψ′ ℓΓxℓ) = Ri(Ψ′ 1,...,Ψ′ ℓ)Γyi, for some yi∈Fn. But both IdGand Ri(Ψ′ 1,...,Ψ′ ℓ) fix tauso, yimust equal ˆuαifor some αi∈Z. Therefore, IdG=Ri(Ψ′ 1,...,Ψ′ ℓ)Γαi ˆu,i= 1,...,d, is a first set of relations for AutH(G). On the other hand, for each generating letter, zj, of Fn,j= 1,...,n, we have the equality Γzj=Sj(Ψ1,...,Ψℓ) = Sj(Ψ′ 1Γx1,...,Ψ′ ℓΓxℓ) = Sj(Ψ′ 1,...,Ψ′ ℓ)Γyj, for some yj∈Fn. But then IdG=Sj(Ψ′ 1,...,Ψ′ ℓ)Γyjz−1 j, which implies yjz−1 j= ˆuβjfor some βj∈Z. Therefore, IdG= Sj(Ψ′ 1,...,Ψ′ ℓ)Γβj ˆu,j= 1,...,n, is a second set of relations for AutH(G). Finally, observe that for k= 1,...,ℓ, ˆuΨ′ k=tckˆufor some ck∈Zmand thus, Γˆucommutes with Ψ′ k. Therefore, Ψ′ kΓˆu= ΓˆuΨ′ k,k= 1,...,ℓ, is a third set of relations for AutH(G). We are going to prove that (7) AutH(G)≃Ψ′ 1,...,Ψ′ ℓ,Γˆu Ri(Ψ′ 1,...,Ψ′ ℓ)Γαi ˆu, Sj(Ψ′ 1,...,Ψ′ ℓ)Γβj ˆu,Ψ′ kΓˆu= ΓˆuΨ′ k i=1,...,d j=1,...,n k=1,...,ℓ . To this goal, denote by Gthe group presented by the presentation on the right hand side, where elements are formal words on the ‘symbols’ {Ψ′ 1,...,Ψ′ ℓ,Γˆu}subject to the relations indicated (we abuse notation, denoting by Ψ′ 1,...,Ψ′ ℓ,Γˆuboth the corresponding symbols in G, and the corresponding automorphisms in AutH(G), the real meaning being always clear from the context). Let us construct a map f: AutH(G)→ G, and a group homomorphism G← G :gsuch that fg = IdAutH(G)and gf = IdG. This will suffice to prove (7) and finish the argument. Define gby sending the symbol Ψ′ kto the automorphism Ψ′ k,k= 1,...,ℓ, and the symbol Γˆu to the automorphism Γˆu; since, as we have proved in the three previous paragraphs, the relations from Gare really satisfied in AutH(G), gdetermines a well defined homomorphism from Gto AutH(G). (For later use, we emphasize the meaning of this: every equality holding symbolically in Gholds also genuinely in AutH(G).) On the other hand, for Ψ ∈AutH(G), define Ψf∈ G as follows: write Ψ ∈AutH(G)⩽AutW(G) as a word on Ψ1,...,Ψℓ, say Ψ = v(Ψ1,...,Ψℓ), compute Ψ = v(Ψ1,...,Ψℓ) = v(Ψ′ 1Γx1,...,ΨℓΓxℓ) = v(Ψ′ 1,...,Ψ′ ℓ)Γy=v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu(in AutH(G) !), where y= ˆuρfor some ρ∈Zsince both Ψ and v(Ψ′ 1,...,Ψ′ ℓ) fix tau; and, finally, define Ψfto be the word v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu∈ G. First, we have to see that fis well defined. That is, take Ψ = w(Ψ1,...,Ψℓ) another expression for Ψ, write Ψ = w(Ψ1,...,Ψℓ) = w(Ψ′ 1,...,Ψ′ ℓ)Γτ ˆu(in AutH(G) !) for the appropriate integer τ∈Z, and we have to prove that the equality v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu=w(Ψ′ 1,...,Ψ′ ℓ)Γτ ˆuholds, abstractly, in G. From the fact v(Ψ1,...,Ψℓ) = Ψ = w(Ψ1,...,Ψℓ) (equalities happening in the group (5)), we deduce that the word v(Ψ1,...,Ψℓ)−1w(Ψ1,...,Ψℓ) is formally a product of conjugates of R1(Ψ1,...,Ψℓ),...,Rd(Ψ1,...,Ψℓ), say v(Ψ1,...,Ψℓ)−1w(Ψ1,...,Ψℓ) = N Y k=1 Rǫk ik(Ψ1,...,Ψℓ)ck(Ψ1,...,Ψℓ). FIXED SUBGROUPS AND AUTO-FIXED CLOSURES IN FREE-ABELIAN TIMES FREE GROUPS 19 Particularizing this identity on Ψ′ 1,...,Ψ′ ℓ∈ G, and working in G(i.e., only using symbolically the defining relations for G), we have that v(Ψ′ 1,...,Ψ′ ℓ)−1w(Ψ′ 1,...,Ψ′ ℓ) = N Y k=1 Rǫk ik(Ψ′ 1,...,Ψ′ ℓ)ck(Ψ′ 1,...,Ψ′ ℓ)= = N Y k=1 Γ−ǫkαik ˆuck(Ψ′ 1,...,Ψ′ ℓ)= N Y k=1 Γ−ǫkαik ˆu= Γ−PN k=1 ǫkαik ˆu. But, applying g(i.e., reading the above equality in AutH(G)), we have IdG=v(Ψ1,...,Ψℓ)−1w(Ψ1, . . . , Ψℓ) = Γ−ρ ˆuv(Ψ′ 1,...,Ψ′ ℓ)−1w(Ψ′ 1,...,Ψ′ ℓ)Γτ ˆu= Γτ−ρ−PN k=1 ǫkαik ˆu and so, the exponent must be zero, τ−ρ−PN k=1 ǫkαik= 0, because n⩾2. Going back to G, we conclude that Γ−ρ ˆuv(Ψ′ 1,...,Ψ′ ℓ)−1w(Ψ′ 1,...,Ψ′ ℓ)Γτ ˆu= Γτ−ρ−PN k=1 ǫkαik ˆu= 1, showing that the map fis well defined. Now consider the composition fg : AutH(G)→ G → AutH(G): for every Ψ ∈AutH(G), write (in AutH(G) !) Ψ = v(Ψ1,...,Ψℓ) = v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu,ρ∈Z, and we have Ψf=v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu∈ G. But then, Ψfg =v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆug=v(Ψ′ 1,...,Ψ′ ℓ)Γρ ˆu= Ψ (in AutH(G) !). Hence, fg = IdAutH(G). Finally, consider the composition gf :G → AutH(G)→ G. Take k= 1,...,ℓ and, in order to compute Ψ′ kgf = Ψ′ kf, we have to express Ψ′ k∈AutH(G) as a word on Ψ1,...,Ψℓ; take, for example, Ψ′ k= ΨkΓ−1 xk= ΨkΓ−1 xk(z1,...,zn)= Ψkxk(Γz1,...,Γzn)−1= Ψkxk(S1(Ψ1,...,Ψℓ),...,Sn(Ψ1,...,Ψℓ))−1; then, rewrite in terms of Ψ′ 1,...,Ψ′ ℓ, Ψ′ k= Ψkxk(S1(Ψ1,...,Ψℓ), . . . , Sn(Ψ1,...,Ψℓ))−1= Ψ′ kxk(S1(Ψ′ 1,...,Ψ′ ℓ),...,Sn(Ψ′ 1,...,Ψ′ ℓ))−1Γρ ˆu, for the appropriate integer ρ∈Z; and we have, in G(i.e., only using symbolically the defining relations for G), Ψ′ kgf = Ψ′ kf= Ψ′ kxk(S1(Ψ′ 1,...,Ψ′ ℓ),...,Sn(Ψ′ 1,...,Ψ′ ℓ))−1Γρ ˆu = Ψ′ kxk(Γ−β1 ˆu,...,Γ−βn ˆu)−1Γρ ˆu = Ψ′ kΓxab kβT ˆuΓρ ˆu = Ψ′ kΓxab kβT+ρ ˆu, where β= (β1,...,βn)∈Zn. But, applying g, using fg = IdAutH(G), and cancelling Ψ′ ifrom the left, we obtain IdG= Γxab kβT+ρ ˆuand so, xab kβT+ρ= 0. Hence, back in G, Ψ′ kgf = Ψ′ k, for k= 1,...,ℓ. Similarly, Γˆugf = Γˆuf=ˆu(Γz1,...,Γzn)f =ˆu(S1(Ψ1,...,Ψℓ),...,Sn(Ψ1,...,Ψℓ))f = ˆu(S1(Ψ′ 1,...,Ψ′ ℓ),...,Sn(Ψ′ 1,...,Ψ′ ℓ))Γρ ˆu = ˆu(Γ−β1 ˆu,...,Γ−βn ˆu)Γρ ˆu = Γ−ˆuabβT+ρ ˆu, for the appropriate integer ρ∈Z. But, applying g, we obtain Γˆu= Γ−ˆuabβT+ρ ˆu(in AutH(G) !) and so, −ˆuabβT+ρ= 1. Hence, back in G, Γˆugf = Γˆu, finishing the proof that gf = IdG. 20 MALLIKA ROY AND ENRIC VENTURA This completes the proof of the isomorphism (7) and so, the proof of the Theorem.  The above proof that stabilizers of subgroups of G=Zm×Fnare finitely presented (and a finite presentation is computable) makes a strong use of the fact that the center of Gis Zm, i.e., the elements of the form tacommute with everybody in G. For this reason, this proof is far from generalizing to arbitrary right angled Artin groups, providing an analog of Day’s Theorem 5.10 for real elements instead of conjugacy classes. This suggests the following question, which is open as far as we know. Question 6.1. 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