Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups

arXiv:2307.07809v2 [math.GR] 9 Mar 2024 UNIQUENESS OF ROOTS UP TO CONJUGACY IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS OWEN GARNIER LAMFA, Universit´e de Picardie Jules Verne, CNRS UMR 7352, 33, rue Saint-Leu, 80000, Amiens, France. Abstract. We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the ∆-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups. Contents 1. Preliminaries on Garside monoids 3 1.1. Definitions, normal form 3 1.2. Conjugacy and periodic elements 4 2. Circular monoids 6 2.1. Definition, first properties 6 2.2. Conjugacy in circular groups 9 2.3. Some group theoretic properties 12 2.4. Application to complex braid groups of rank 2 15 3. ∆-product and hosohedral-type groups 16 3.1. ∆-product of Garside monoids 16 3.2. Hosohedral-type Garside groups 21 References 24 Introduction The aim of this article is to study two particular classes of Garside monoids, called circular and hosohedral-type monoids, along with their associated Garside groups. Circular monoids were first defined in the original article of Dehornoy and Paris where the authors introduce the notion of Garside group. Hosohedral-type monoids first appeared in the work of Matthieu Picantin ([Pic00]) and later on in that of Mireille Soergel ([Soe23]) because of the particular properties of their lattices of simples. Let m, ℓ be two positive integers. The circular monoid M(m, ℓ) is defined by a monoid presentation, given by a set of mordered generators, endowed with relations stating that any product of ℓconsecutive generators should be equal (Definition 2.1). The circular E-mail address:[email protected]. Date: March 12, 2024. 2020 Mathematics Subject Classification. Primary 20F36 20F05, Secondary 20E06. Key words and phrases. Garside monoid, Garside group, Complex braid groups. 1 2 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS group G(m, ℓ) is then defined as the enveloping group of M(m, ℓ). Circular monoids are Garside monoids, with Garside element the product of ℓconsecutive generators. Our main motivation is to study roots in circular groups using their Garside structure, in particular, the associated solution to the conjugacy problem. We reduce the situation to the study of two classes of elements, namely rigid and periodic elements (Proposition 2.7). Generally speaking, let (M, ∆) be a Garside monoid, and let p, q be integers. An element ρin the associated Garside group G(M) is said to be (p, q)-periodic if ρp= ∆q. We give a complete description of periodic elements in circular groups, up to conjugacy. Proposition. (Proposition 2.13 and Corollary 2.11) Let m, ℓ be two positive integers. An element x∈G(m, ℓ)is periodic if and only if it is conjugate to a power of a product of either mor ℓconsecutive generators. In particular, for a given pand q, all (p, q)-periodic elements are conjugate. Moreover, an element x∈G(m, ℓ)is periodic if and only if it has a central power. As a consequence, an isomorphism between two circular groups must preserve periodic elements. From this we deduce the classification of circular groups up to isomorphism. Corollary. (Corollary 2.21) Let m, ℓ, m′, ℓ′be four positive integers. The groups G(m, ℓ) and G(m′, ℓ′)are isomorphic if and only if one of the following holds -1∈ {m, ℓ}and 1∈ {m′, ℓ′}. In this case, G(m, ℓ)≃G(m′, ℓ′)≃Z. -(m′, ℓ′)∈ {(m, ℓ),(ℓ, m)}. Since periodic elements are (by definition) roots of powers of ∆, the above proposition is a first result of uniqueness of roots up to conjugacy in circular groups. By also studying the conjugacy of rigid elements, we obtain the following stronger result. Theorem. (Theorem 2.16) Let m, ℓ be positive integers. If α, β ∈G(m, ℓ)are such that αn=βnfor some nonzero integer n, then αand βare conjugate in G(m, ℓ). We then apply this result to complex braid groups of rank 2. Recall that, if W⊂GLn(C) is a finite group generated by (pseudo-)reflections, then one can consider the braid group B(W), defined as the fundamental group of the regular orbit space associated to W(see [BMR98, Section 2.B]). These groups generalize Artin groups of spherical type. The question of uniqueness of roots up to conjugacy in braid groups was first studied by Juan Gonz´alez-Meneses in [Gon03] , where he proved that roots were unique up to conjugacy in the Artin groups of type A(i.e. the usual braid groups). His results were later expanded in [LL10] to the Artin groups of type B. To our knowledge, no other results are known regarding this question. The uniqueness of roots up to conjugacy is conjectured to hold at least for every spherical Artin group (cf. [DDGKM, Conjecture X.3.10]). The above theorem on uniqueness of roots up to conjugacy in circular groups, along with the fact that every complex braid group of rank 2 is isomorphic to a circular group (see [Ban76, Theorem 1 and Theorem 2]), gives the uniqueness of roots up to conjugacy in complex braid groups of rank 2. Theorem. (Theorem 2.23) Let Wbe an irreducible complex reflection group of rank 2, and let B(W)be its braid group. If α, β ∈B(W)are such that αn=βnfor some nonzero integer n, then αand βare conjugate in B(W). This theorem gives preliminary evidences that the uniqueness of roots up to conjugacy may actually hold in all complex braid groups. In the last section, we study a generalization of circular monoids, called hosohedral-type monoids. These monoid appear as a particular case of a general construction involving Garside monoids, called the ∆-product. Let (M1,∆1),...,(Mh,∆h) be a family of Garside monoids. We define the ∆-product of (M1,∆1)∗∆· · · ∗∆(Mh,∆h) as the quotient of the free product of the Miby the relations UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 3 stating that all the ∆iare equal to each other. This monoid is again a Garside monoid (with Garside element the image of any ∆i), in which each one of the Miembeds. Proposition. (Proposition 3.9) Let (M1,∆1),...,(Mh,∆h)be homogeneous Garside monoids. Each Garside group G(Mi)naturally embeds in the Garside group G(M1∗∆· · · ∗∆Mh). These embeddings respect left-weighted factorizations. From this proposition, we deduce that the center of a ∆-product of Garside monoids (with more than one factor) is cyclic and generated by a power of the Garside element (Proposition 3.12). We also deduce that, for integers p, q, the ∆-product M1∗∆· · · ∗∆Mh admits (p, q)-periodic elements if and only if one of the factors does (Proposition 3.11). Note that two periodic elements coming from two different factors G(Mi) and G(Mj) may not be conjugate in the ∆-product (Example 3.19). We then apply these results to hosohedral-type monoids, which are defined as the ∆- product of circular groups. In this case, there can be distinct conjugacy classes of (p, q)- periodic elements (see Example 3.19). We show that these are actually the only obstructions to the uniqueness of roots up to conjugacy. Theorem. (Theorem 3.21) Let Mbe a hosohedral-type monoid. If α, β ∈G(M)are such that αn=βnfor some nonzero integer n, then we either have •αand βare conjugate. •αand βare nonconjugate periodic elements of G(M). Acknowledgments. This work is part of my PhD thesis, done under the supervision of Pr. Ivan Marin. I thank him for his precious advice, especially in Sections 2.3.1 and 2.3.2. I would also like to thank Mireille Soergel and Igor Haladjian for stimulating discussions. 1. Preliminaries on Garside monoids Throughout this article, the gcd (resp. lcm) of two integers pand qwill be denoted by p∧q(resp. p∨q). Our main arguments rely on the study of super-summit sets and periodic elements. For this we need some reminders on Garside monoids. Our main reference is [DDGKM, Section I.2 and Chapter VIII]. 1.1. Definitions, normal form. We start by considering a monoid M. Throughout this paper, we will always assume that Mis homogeneous. That is, there is some monoid morphism ℓ:M→(Z⩾0,+) such that elements of positive length generate M. This condition is far from minimal when considering Garside monoids, but it is sufficient here as both circular monoids and hosohedral-type monoids are homogeneous. The homogeneity condition implies in particular that the set of invertible elements of Mis trivial. We also assume that Mis cancellative. That is, every equality of the form abc =ab′cin M implies b=b′. Under these assumptions, we define two partial orders and on Mby ab⇔ ∃c∈M|ac =band ba⇔ ∃c∈M|b=ca. These partial order are called left-divisibility and right-divisibility, respectively. A nontrivial element a∈Mis an atom if it admits no nontrivial (right- or left-)divisors other than itself. We say that an element ∆ ∈Mis balanced if its sets of left- and right-divisors are equal, we then simply call this set the divisors of ∆. Definition 1.1. [DDGKM, Definition I.2.1] Let Mbe a homogeneous cancellative monoid, such that the posets (M, ) and (M, ) are both lattices. A Garside element in Mis a balanced element ∆ whose set of divisors S is finite and generates M. We call Sthe set of simple elements (associated to ∆) and we say that (M, ∆) is a homogeneous Garside monoid. 4 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS The assumption that (M, ) and (M, ) are both lattices means that (left- and right-) lcms and gcds of two elements always exist. We denote by a∧b(resp. a∨b) the left-gcd (resp. right-lcm) of two elements a, b. If need be, we will denote by ∧Rthe right-gcd and by ∨Lthe left-lcm. Let (M, ∆) be a homogeneous Garside monoid, and let s∈Sbe a simple element. By cancellativity, there is a unique simple ssuch that ss = ∆. We call sthe left-complement of sin ∆. Likewise, the right-complement s∗of sin ∆ is defined by s∗s= ∆. Defining φ(s) := sfor any simple element syields an automorphism of M, which permutes the simple elements (in particular, it has finite order). We call φthe Garside automorphism of (M, ∆). By definition, we have s∆ = ∆φ(s) for any simple element s. One of the main features of a Garside monoid is that it gives rise to a convenient solution to the word problem, given by the notion of greedy word. Definition 1.2. [DDGKM, Corollary V.1.54] Let (M, ∆) be a homogeneous Garside monoid. A word st of length 2 in Sis called greedy if s= (st)∧∆, or equivalently if sand tare left-coprime. In general, a word s1···srin S is called greedy if each subword sisi+1 is greedy for i∈[[1, r −1]]. Lemma 1.3. Let s, t be two simple elements in a homogeneous Garside monoid (M, ∆), and let u:= t∧s. The greedy normal form of the product st is given by s′t′where s′=su ∈Sand t′=u−1t∈S. Proof. This is an application of the invariance of left-gcd under multiplication, as we have su =s(t∧s) = (st)∧(ss) = x∧∆.  An immediate induction on this lemma shows that any element in a homogeneous Garside monoid Madmits a unique decomposition as a greedy word in the simple elements. The lemma also gives a practical way to compute the greedy decomposition of any element xof M, by the means of successively computing greedy decompositions of pairs of elements of S(see for instance [DDGKM, Algorithm III.1.52]). The definition of a homogeneous Garside monoid implies the Ore condition, thus a homogeneous Garside monoid Malways embeds in its enveloping group G(M), which can be conveniently described as a group of fractions. The greedy decomposition in Mcarries on into a complete description of the elements of G(M). Proposition 1.4. [DDGKM, Proposition I.2.4] Let (M, ∆) be a homogeneous Garside monoid. Every element x∈G(M)admits a unique decomposition of the form x= ∆kbwith k∈Z, b ∈Mand ∆6 b. In particular, this proposition gives a solution to the word problem in G(M), given by computing the above decomposition, and then the greedy decomposition of bin M. This decomposition of an element x∈G(M) is the left-weighted factorization of x. 1.2. Conjugacy and periodic elements. In this section, we fix (M, ∆) a homogeneous Garside monoid, and G(M) its enveloping group. The Garside monoid Mprovides a solution to the conjugacy problem in G(M), given by the computation of super-summit sets. These super-summit sets also allow for the computation of centralizers in G(M). Let x∈G(M) with left-weighted factorization x= ∆ks1···sr. The infimum (resp. the supremum) of xis defined by inf(x) := k(resp. sup(x) = k+r). Definition 1.5. [DDGKM, Defintion VIII.2.12] Let x∈G(M). The conjugacy class of xin G(M) admits a well-defined subset SSS(x), on which each one of inf and sup takes a constant value. Furthermore, for every conjugate x′ of xin G(M), we have inf(x)⩽inf(SSS(x)) and sup(x′)⩾sup(SSS(x)). The set SSS(x) is called the super-summit set of x. UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 5 Since the set Sis finite, there is a finite number of elements of G(M) with given inf and sup. In particular, the super-summit set of any element of G(M) is always finite. As the super-summit set of an element depends only on its conjugacy class, computing this set gives a solution to the conjugacy problem. To compute super-summit sets, one uses the cycling and decycling operations. Let x∈G(M) with left-weighted factorization x= ∆ks1···sr. The initial factor (resp. final factor) of xis defined as init(x) := φ−k(s1) (resp. fin(x) := sr). The cycling (resp. decycling) of xis then defined as cyc(x) = xinit(x)=xφ−k(s1)(resp. dec(x) = xfin(x)−1=xs−1 r). Proposition 1.6. [DDGKM, Proposition VII.2.16] Let x∈G(M). One can go from xto an element of SSS(x)by a finite sequence of cycling, followed by a finite sequence of decycling. This proposition can be used to compute an element of SSS(x) starting from x∈G(M). The whole super-summit set of x∈G(M) can then be encoded in a so-called conjugacy graph. Definition 1.7. [FG03, Section 3] Let x∈G(M), the conjugacy graph CG(x) of xis an oriented graph, defined as follows. •The object set of CG(x) is SSS(x). •An arrow a→bin CG(x) is given by a simple element s∈Msuch that as=band that no left-divisor tof sis such that at∈CG(x). By [DDGKM, Lemma VIII.2.19], the conjugacy graph of any element xis connected. In other words, two elements of SSS(x) are always conjugate by a sequence of simple elements (and their inverses) such that the result at each step remains in SSS(x). Let x∈G(M). By definition of the conjugacy graph, a non-oriented path from xto itself in CG(x) induces a word in the simple elements (and their inverses) representing an element of the centralizer CG(M)(x). By [FG03, Theorem 3.4], the centralizer CG(M)(x) of xis generated by the images in G(M) of any generating set of paths from xto itself in CG(x). In our study of circular and hosohedral-type monoids, we are going to consider two classes of elements showing two extreme behavior with respect to the cycling and decycling operations. Definition 1.8. [BGG07, Definition 3.1] An element x∈G(M), with left-weighted factorization x= ∆ks1···sr, is said to be rigid if r= 0 or if the word fin(x) init(x) is greedy. If x= ∆ks1···sr∈G(M) is rigid (with r > 0), the left-weighted factorizations of cyc(x) and dec(x) are given by cyc(x) = ∆ks2···srφ−k(s1) and dec(x) = ∆kφk(sr)s1···sr−1. In particular, inf(cyc(x)) = inf(x) = inf(dec(x)) and sup(cyc(x)) = sup(x) = sup(dec(x)). As both cyc(x) and dec(x) are also rigid, we obtain that a rigid element always belongs to its super-summit set by Proposition 1.6. Rigid elements also exhibit a strict behavior with respect to powers. This behavior is useful in relating the super-summit set of a rigid element to that of its powers. Lemma 1.9. Let (M, ∆) be a Garside monoid, and let α, β ∈G(M)be rigid elements. For any positive integer n, we have cyc(α)n= cyc(αn),dec(α)n= dec(αn)and φ(αn) = φ(α)n. Moreover, if αn=βnfor some positive integer n, then α=β. Proof. The result is trivial if n= 1, we assume that n⩾2 from now on. Let α= ∆ks1···sr be the left-weighted factorization of α. Since αis rigid, the left-weighted factorization of 6 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS x:= αnis αn= ∆nkφ(n−1)k(s1···sr)···φk(s1···sr)s1···sr. We note that the initial factor (resp. the final factor) of αnis the same as that of α, hence the result on cyc(αn) and dec(αn). The result on φ(αn) is an obvious consequence of φ(x) = x∆for x∈G(M). Moreover, we have inf(αn) = ninf(α) and sup(αn) = nsup(α). The left-weighted factorization of αcan be recovered using that of αnby taking ∆inf(αn) n, followed by the last sup(αn) nterms of the left-weighted factorization of αn. Since this depends only on the leftweighted factorization of αn, we obtain that αn=βnimplies α=βif βis another rigid element.  The other class of elements we will be interested in is that of periodic elements. Definition 1.10. [DDGKM, Definition V.3.2] Let (M, ∆) be a homogeneous Garside monoid, and let p, q be two integers. An element ρ∈G(M) is said to be (p, q)-periodic if ρp= ∆q. By definition, any root and any power of a periodic element in G(M) is again a periodic element. Moreover, it is obvious by definition that a (p, q)-periodic element is also (np, nq)- periodic for any integer n⩾1. One can show that the converse is also true: Proposition 1.11. [DDGKM, Proposition VIII.3.31 and Proposition VIII.3.34] Let p, q be two integers. Let d:= p∧q, with p=dp′and q=dq′. If p′= 1, then any (p, q)-periodic element in G(M)is conjugate to ∆q′. If p′6= 1, then any (p, q)-periodic element in G(M)is conjugate to a (p′, q′)-periodic element of the form ∆ks, where s∈S. In particular, this proposition shows that the super-summit set of a periodic element contains only elements of the form ∆ksor ∆k(as such elements clearly lie in their own super-summit set). The definition of rigid element is a priori not group-theoretic and depends on the Garside monoid M. On the contrary, periodic elements can sometimes be characterized by a solely group-theoretic property. If (M, ∆) is a homogeneous Garside monoid such that the center of G(M) is cyclic and generated by a power of ∆, then an element ρ∈G(M) is periodic if and only if it admits a central power. In this case, we can try and compare two groups by comparing their respective periodic elements. Definition 1.12. Let (M, ∆) be a homogeneous Garside monoid. We say that a periodic element ρ∈G(M) is irreducible if it admits no roots in G(M) other than itself. Since (M, ∆) is assumed to be homogeneous here, we obtain that any periodic element in G(M) is always a power of some irreducible periodic elements. Let (M, ∆),(M′,∆′) be two homogeneous Garside monoids, such that the center of G(M) (resp. of G(M′)) is cyclic and generated by some power of ∆ (resp. ∆′). An isomorphism G(M)→G(M′) sends a generator of Z(G(M)) to a generator of Z(G(M′)). Therefore, it must map (irreducible) periodic elements of G(M) to (irreducible) periodic elements of G(M′). 2. Circular monoids In this section we define circular monoids by their (monoid) presentations. These monoids already appeared in [DP99, Example 5], where the authors showed that they were Garside monoids. Here we propose an in-depth study of their Garside properties. 2.1. Definition, first properties. Let m, ℓ be two positive integers that we fix throughout this section. Let also {a0,...,am−1}be an alphabet. For i∈Zand p∈Z⩾1, we define s(i, p) as the word s(i, p) := p Y k=i′ ak=ai′ai′+1 ···ai′+p−1 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 7 where i′is the remainder in the Euclidean division of iby m, and with the convention that, for j⩾1, aj:= aj′where j′is the remainder in the Euclidean division of jby m. We also define s(i, 0) to be the empty word for all i∈Z. Definition 2.1. Let m, ℓ be two positive integers. The circular monoid M(m, ℓ) is defined by the monoid presentation M(m, ℓ) := ha0,...,am−1| ∀i[[0, m −1]], s(i, ℓ) = s(i+ 1, ℓ)i+. The enveloping group G(m, ℓ) of M(m, ℓ) is called a circular group. From now on, we assimilate the word s(i, p) (i∈Z,p⩾0) with its image in M(m, ℓ). Example 2.2.The monoid M(3,3) is given by ha, b, c |abc =bca =cabi+. The group G(3,3) is the fundamental group of the complement of 3 lines going through the origin in C2(cf. [DP99, Example 5]). The monoid M(2,3) is given by hs, t |sts =tsti+. It is the Artin monoid of type A2. We note that the presentation of M(m, ℓ) is homogeneous. That is, the defining relations are equalities between words of the same length. The function sending an element of M(m, ℓ) to the length of any word representing it is then a length function, making M(m, ℓ) into a homogeneous monoid. As the only defining relations of M(m, ℓ) are between words of length ℓ, two words of length less than ℓcannot represent the same element of M(m, ℓ). In particular, for 0 < p < ℓ and i, i′∈Z, we have s(i, p) = s(i′, p) in M(m, ℓ) if and only if i≡i′[m]. Lemma 2.3. [DP99, Example 5] The monoid M(m, ℓ)is a homogeneous Garside monoid with Garside element ∆ = s(0, ℓ). Its simple elements are the s(i, p)for i∈[[0, m −1]] and p∈[[0, ℓ]]. The Garside automorphism of M(m, ℓ)is given by φ(s(i, p)) = s(i+ℓ, p). The only statement which is not showed in [DP99, Example 5] is the statement on the Garside automorphism, which comes from the fact that, for any simple element s(i, p) of M(m, ℓ), we have s(i, p)∆ = s(i, p)s(i+p, ℓ) = s(i, ℓ +p) = s(i, ℓ)s(i+ℓ, p) = ∆s(i+ℓ, p). Let s(i, p) be a simple element of M(m, ℓ). We have s(i, p)s(i+p, ℓ −p) = s(i, ℓ) = ∆, thus the right-complement (resp. left-complement) of s(i, p) in ∆ is given by s(i, p)∗=s(i+p−ℓ, ℓ −p) (resp. s(i, p) = s(i+p, ℓ −p)). The fact that any simple element different from 1 and ∆ admits a unique decomposition as a product of atoms has the following consequences. Lemma 2.4. (Left-gcd of two simple elements) Let s(i, p)and s(i′, p′)be two simple elements of M(m, ℓ). (a) We have s(i, p)s(i′, p′)if and only if i=i′and p⩽p′, or if p= 0, or if p′=ℓ. (b) The left-gcd of s(i, p)and s(i′, p′)is given by s(i, p)∧s(i′, p′) =      s(i, p)if s(i, p)s(i′, p′) s(i′, p′)if s(i′, p′)s(i, p) 1otherwise (c) The right-lcm of s(i, p)and s(i′, p′)is given by s(i, p)∨s(i′, p′) =      s(i′, p′)if s(i, p)s(i′, p′) s(i, p)if s(i′, p′)s(i, p) ∆otherwise 8 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS Proof. (a) The cases p= 0 and p′=ℓare immediate: they give s(i, p) = 1 and s(i′, p′) = ∆, respectively. If i=i′and p⩽p′, we have s(i, p)s(i+p, p′−p) = s(i′, p′) and s(i, p)s(i′, p′). Conversely, suppose that s(i, p)s(i′, p′). Since we assume that p′< ℓ and 0 < p, there is only one way to write both s(i, p) and s(i′, p′) as a product of atoms. Thus the assumption that s(i, p) can be written as a prefix of some word expressing s(i′, p′) implies that s(i, p) is in fact the only prefix of length pof s(i′, p′). We obtain that p⩽p′and s(i′, p) = s(i, p), which gives i=i′. (b) The first two cases are obvious. Suppose that we have s(i, p)6 s(i′, p′) and s(i′, p′)6 s(i, p). By the first point we have p, p′/∈ {0, ℓ}and i6=i′. Again by point (a), a nontrivial common left-divisor s(j, q) of s(i, p) and s(i′, p′) should be such that j=i=i′, which is impossible. We apply similar reasoning to prove point (c).  This very strict behavior of gcds and lcms allows for an easy description of greedy normal forms of a product of two simple elements in M(m, ℓ). Lemma 2.5. (Greedy normal form of a product of two nontrivial simples) Let s(i, p)and s(i′, p′)be two simple elements of M(m, ℓ)with 0< p, p′< ℓ. The greedy normal form of s(i, p)s(i′, p′)is given by s(i, p)s(i′, p′) =          s(i, p)s(i′, p′)if i+p6≡ i′[m], s(i, p +p′)if i+p≡i′[m]and p+p′< ℓ, ∆if i+p≡i′[m]and p+p′=ℓ, ∆s(i+ℓ, p +p′−ℓ)if i+p≡i′[m]and p+p′> ℓ. Proof. We will apply Lemma 1.3. Since 0 < p, p′< ℓ, Lemma 2.4 gives s(i, p)∧s(i′, p′) = s(i+p, ℓ −p)∧s(i′, p′), =     s(i+p, ℓ −p) if i+p≡i′[m] and ℓ−p⩽p, s(i′, p′) if i+p≡i′[m] and p′⩽ℓ−p 1 otherwise. These three cases give the desired result.  This particular description of greedy normal forms in circular monoids will induce a convenient description of super-summit sets in Section 2.2. We finish this section by the study of the particular case where m=ℓ. In this case, the group G(m, m) is the fundamental group of the complement of mlines through the origin in C2(cf. [DP99, Example 5]). Its presentation can be reinterpreted as a direct product. Lemma 2.6. Let mbe a positive integer, and let Fm−1be a free group with m−1generators. The group G(m, m)is isomorphic to Z×Fm−1. This isomorphism identifies ∆with (z, 1), where zis a generator of Z. Proof. The result is obvious if m∈ {1,2}. We denote by x0,··· , xm−2the generators of Fm−1, and by za generator of Z. We define f:Z×Fm−1→G(m, m) by (f(z) = ∆ = s(0, m), f(xi) = ai∀i∈[[0, m −2]]. This induces a well-defined morphism since ∆ ∈Z(G(m, m)). Conversely, we define g: G(m, m)→Z×Fm−1by (g(ai) = xi∀i∈[[1, m −2]], g(am−1) = (x0···xm−2)−1z. UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 9 This induces a well-defined morphism since, for all i∈[[0, m −1]], we have g(s(i, m)) = g(s(i, m −1−i)am−1s(0, i)) =xi···xm−2(x0···xm−2)−1zx0···xi−1 =xi···xm−2(x0···xm−2)−1x0···xi−1z=z=g(s(0, m)). It is straightforward to check that fand gare inverses of one another.  2.2. Conjugacy in circular groups. Again we fix two positive integers m, ℓ. In this section, we study separately the conjugacy of periodic and non-periodic elements in circular groups. Our first result shows that one can reduce to the case of periodic or rigid elements. Proposition 2.7. Let xbe an element of G(m, ℓ). If xlies in its own super-summit set, then it is either rigid or periodic. Proof. First, if inf(x) = sup(x), then we have x= ∆kfor some k∈Z. In this case, xis obviously both rigid and (1, k)-periodic. Suppose now that sup(x) = inf(x) + 1. We have x= ∆ks(i, p) for some i∈[[0, m −1]] and p∈[[1, ℓ −1]]. The element xis rigid if and only if the word s(i, p)φ−k(s(i, p)) = s(i, p)s(i−kℓ, p) is greedy. By Lemma 2.5, this is equivalent to i+p≡i−kℓ[m], i.e. kℓ +p≡0[m]. If kℓ +p≡0[m], then there is some integer vwith −kℓ +mv =p. In particular we have ℓ∧m|p. We have xn= ∆knφk(n−1)(s(i, p)) ···φk(s(i, p))s(i, p) = ∆kns(i+k(n−1)ℓ, p)···s(i+kℓ, p)s(i, p) = ∆kns(i+k(n−1)ℓ, np) = ∆kn+as(i+ (k(n−1) + a)ℓ, r), where np =aℓ +ris the Euclidean division of np by ℓ. The remainder ris 0 if and only if nis a multiple of ℓ∨p p=ℓ ℓ∧p. We obtain that xis (a, b)-periodic, for a=ℓ ℓ∧pand b=kℓ∨p p+ℓ∨p ℓ=kℓ +p ℓ∧p=mv ℓ∧p. Furthermore, aand bare coprime. Lastly, suppose that sup(x)>inf(x) + 1. The left-weighted factorization of xis given by ∆ks1···srwith r > 1. We claim that xis rigid. Otherwise, the word srφ−k(s1) is not greedy. By Lemma 2.5, we either have that srφ−k(s1) is a simple element, or a product of the form ∆swhere sis a simple element different from 1 and ∆. In the first case, we have sup(cyc(x)) <sup(x). In the second case, we have inf(cyc(x)) >inf(x). In both cases, we have x /∈SSS(x).  2.2.1. Periodic elements. The proof of Proposition 2.7 also gives the following result. Lemma 2.8. Let m, ℓ be positive integers, and let ∆ks(i, p)be in G(m, ℓ)with 0< p < ℓ. The element ∆ks(i, p)is periodic if and only if p+kℓ ≡0[m], in which case it is a (ℓ p∧ℓ,mv p∧ℓ)-periodic element (where p+kℓ =mv). Now that we have a characterization of the elements of the super-summit sets of periodic elements, we can compute conjugacy graphs and centralizers. We distinguish two cases. Lemma 2.9. Let x= ∆kfor some nonzero integer k. We have SSS(∆k) = {∆k}. The centralizer of ∆kin G(m, ℓ)is either G(m, ℓ)if kℓ is a multiple of m, or cyclic and generated by ∆otherwise. Proof. We have that y∈G(m, ℓ) lies in SSS(∆k) only if inf(y) = k= sup(y). The only element satisfying this is ∆k, which is conjugate to itself. Thus we have SSS(∆k) = {∆k}. 16 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS This is mostly a rephrasing of [Ban76, Theorem 1 and Theorem 2]: - The only non-irreducible cases are groups of the form W=Z/dZ/×Z/d′Z(with d, d′⩾1). In this case we have B(W)≃Z×Z≃G(2,2). - If W=G(de, e, 2) for eodd and d⩾2 or W∈ {G5, G10, G18}, then B(W)≃G(2,4). - If W=G(de, e, 2) for eeven and d⩾2 or W∈ {G7, G11, G15, G19}, then B(W)≃G(3,3). - If W=G(e, e, 2) for e⩾3, then B(W)≃G(2, e). - If W∈ {G4, G8, G16}, then B(W)≃G(2,3). - If W∈ {G6, G9, G13, G17}, then B(W)≃G(2,6). - If W=G14, then B(W)≃G(2,8). - If W=G20, then B(W)≃G(2,5). - If W=G21, then B(W)≃G(2,10). - If W=G12, then B(W)≃G(3,4). - If W=G22, then B(W)≃G(3,5). A direct application of Theorem 2.16 then gives Theorem 2.23. Let Wbe a complex reflection group of rank 2, and let B(W)be its braid group. If α, β ∈B(W)are such that αn=βnfor some nonzero integer n, then αand β are conjugate in B(W). Remark 2.24.Our approach only covers complex reflection groups of rank 2. Indeed, by Lemma 2.17, circular groups have homological dimension at most 2, and a complex braid group of rank rhas homological dimension rby [CM14, Proposition 1.1]. Thus only rank 2 complex reflection groups can have a braid group isomorphic to a circular group. 3. ∆-product and hosohedral-type groups In this section we present hosohedral-type Garside groups as a generalization of circular groups. These groups are enveloping groups of so-called hosohedral-type monoids. These monoids were first introduced by Picantin in his PhD thesis under the name “mono¨ıdes de type fuseau” ([Pic00, D´efinition 1.3]). In [Pic00, Proposition 2.4], Picantin shows that these monoids are exactly the Garside monoids whose lattice of simples has the shape of a hosohedron (“fuseau” in french). The name “hosohedral-type monoid” was suggested to us by Picantin. More recently, hosohedral-type groups were identified by Mireille Soergel in [Soe23, Theorem 4.6] as the Garside groups satisfying a particular nonpositive curvature property (namely, the systolicity of the flag complex associated to the “Garside presentation” as in [Soe23, Lemma 4.3]). Here we introduce these groups as a particular case of a general construction, already present in [DP99], which we call the ∆-product of Garside monoids. 3.1. ∆-product of Garside monoids. Let (M1,∆1),...,(Mh,∆h) be a finite family of homogeneous Garside monoids that we fix throughout this section. We denote by ℓ1,...,ℓh the associated length functions, and by S1,...,Shthe associated set of simples. We also set A1,...,Ahthe set of atoms of M1,...,Mh, respectively. The free product M1∗ · · · ∗ Mhis not a Garside monoid because two atoms coming from a different factor Mido not have any common multiple (in particular, no lcms). We can fix this by forcing the ∆ito all be equal. Definition 3.1. Let (M1,∆1),...,(Mh,∆h) be a family of homogeneous Garside monoids. The ∆-product of the Miis defined by M1∗∆M2∗∆· · · ∗∆Mh:= ∗h i=1Mi.(∆i= ∆j∀i, j ∈[[1, h]]) Likewise, we define the ∆-product of the enveloping groups G(Mi) by G(M1)∗∆G(M2)∗∆· · · ∗∆G(Mh) = ∗h i=1G(Mi).(∆i= ∆j∀i, j ∈[[1, h]]) UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 17 Remark 3.2.The definition of the ∆-product of Garside monoids depends really on the Garside element and not only on the monoid themselves. For instance, we have M(1, ℓ)≃ Z⩾0as a monoid for every ℓ⩾1. However, we have M(1, p)∗∆M(1, q)≃ ha, b |ap=bqi+. If p, q ⩾2, then this monoid has two atoms and cannot be isomorphic to Z⩾0≃M(1,1)∗∆ M(1,1). Furthermore, the enveloping group of this monoid is not always isomorphic to Z=G(M(1,1) ∗∆M(1,1)). Remark 3.3.Note that, if (M, ∆) is a Garside monoid, the monoid M∗∆M(1,1) is naturally isomorphic to M. Thus we can assume that all the (Mi,∆i) are distinct from M(1,1). We first show that the enveloping group of the ∆-product of the (Mi,∆i) identifies with the ∆-product of the enveloping groups G(Mi). Lemma 3.4. Let (M1,∆1),...,(Mh,∆h)be a family of homogeneous Garside monoids. Let also Mbe the ∆-product M1∗∆· · ·∗∆Mh. The groups G(M)and G(M1)∗∆· · ·∗∆G(Mh) are naturally isomorphic. Proof. If M, M′are two monoids, we denote by Hom(M, M′) the set of monoid morphisms from Mto M′. If Gand G′are two groups, Hom(G, G′) is in fact the set of group morphisms from Gto G′. Let Hbe a group. By definition of the enveloping group and of the ∆-product, we have natural bijections Hom(G(M), H)≃Hom(M1∗∆· · · ∗∆Mh, H) ≃ {f∈Hom(M1∗ · · · ∗ Mh, H)| ∀i, j ∈[[1, h]], f(∆i) = f(∆j)} ≃((fi)∈ n Y i=1 Hom(Mi, H)| ∀i, j, fi(∆i) = fj(∆j)) ≃((fi)∈ n Y i=1 Hom(G(Mi), H)| ∀i, j, fi(∆i) = fj(∆j)) ≃ {f∈Hom(G(M1)∗ · · · ∗ G(Mh), H)| ∀i, j ∈[[1, h]], f(∆i) = f(∆j)} ≃Hom(G(M1)∗∆· · · ∗∆G(Mh), H) Applying this to H:= G(M) gives a bijection Hom(G(M), G(M)) ≃Hom(G(M1)∗∆· · ·∗∆ G(Mh), G(M)). The image of the identity morphism of G(M) under this bijection gives the desired result.  From now on, we will identify G(M1∗∆· · · ∗∆Mh) with G(M1)∗∆· · · ∗∆G(Mh). We now fix a family (M1,∆1),...,(Mh,∆h) of Garside monoids, all distinct from M(1,1). We denote by Mthe associated ∆-product, and by Aits set of atoms. Let i∈[[1, h]]. By definition of the ∆-product as a quotient of the free product, there is a natural morphism ϕi:G(Mi)→G(M). We also denote by ϕithe restriction from Mito M. Proposition 3.5. [DP99, Proposition 5.3] The atoms of the monoid Mare given by A:= Fh i=1 ϕi(Ai). Furthermore, (M, ∆) is a homogeneous Garside monoid with simple elements S= h [ i=1 ϕi(Si). Where ∆ = ϕi(∆i)for any i∈[[1, h]]. The Garside automorphism φof (M, ∆) is given on a simple ϕi(s)by φ(ϕi(s)) = ϕi(φi(s)), where φiis the Garside automorphism of (Mi,∆i). The assertion on the atoms and the assertion that (M, ∆) is a homogeneous Garside monoid are in the original statement of [DP99, Proposition 5.3]. For i∈[[1, h]], we identify 18 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS Aiwith the subset ϕi(Ai) of Afrom now on. The proof of [DP99, Proposition 5.3] can also be used to show the assertion on the simple elements: First, let i∈[[1, h]] and let s∈Sibe a simple element of (Mi,∆i). We have ss= ∆iand ϕi(s)ϕi(s) = ∆. Thus ϕi(s) is simple and ϕi(s) = ϕi(s). We also obtain that the Garside automorphism is given by φ(ϕi(s)) = ϕi(s) = ϕi(φi(s)). Conversely, we have to show that, if s∈Sis a simple element of (M, ∆), then there is some i∈[[1, h]] and some simple es∈Si with s=ϕi(es). This is a direct consequence of the following lemma. Lemma 3.6. Let wbe a word in Awhich expresses a simple element s∈S. There is some i∈[[1, h]] such that all the letters of wlie inside ϕi(Ai). The word wthen also represents some es∈Siwith ϕi(es) = s. Furthermore, if s /∈ {1,∆}, then the integer iand the simple es∈Siare unique. Proof. Let i∈[[1, h]]. For each pair a, b ∈Ai, we choose two words fi(a, b) and fi(b, a) such that afi(a, b) and bfi(b, a) are two words in Aiexpressing a∨bin Mi. By [DP99, Theorem 4.1], the monoid Miadmits the following presentation Mi=hafi(a, b) = bfi(b, a)|a, b ∈Aii+. Now, for a∈Ai, we choose a word c(a) in Airepresenting ain Mi. The proof of [DP99, Proposition 5.3] gives that Madmits the presentation (1) M≃*af(a, b) = bf(b, a)f(a, b) = (fi(a, b) if a, b ∈Ai c(a) if a∈Ai, b ∈Aj, i 6=j++ . By definition, if w1=w2is a relation in this presentation, then we either have - There is an i∈[[1, h]] such that all the letters of both w1and w2lie in Ai. In this case, w1=w2also holds in Mi. - There are two distinct integers i, j ∈[[1, h]] and two atoms a∈Ai, b ∈Ajsuch that w1=ac(a) and w2=bc(b). In this case, w1(resp. w2) represents ∆iin Mi(resp. ∆jin Mj). Let wbe a word in Awhich represents ∆ in M, and let a∈Abe an atom of M. By assumption, there is a sequence of words w1,...,wmin Asuch that w1=w,wm=ac(a) and each wkis equivalent to wk+1 by the use of one relation of presentation (1) for k∈[[1, m−1]]. Up to changing the atom a, we can assume that mis the first integer such that wmis equal to a word of the form bc(b) for some atom b. Let i∈[[1, h]] be such that a∈Ai. We claim that for all j∈[[1, m]], the word wjcontains letters only in Ai. If this is not the case, then let k0be the last integer in [[1, m]] such that wk0contains letters not lying in ϕi(Ai) (we have k0< m by assumption). The defining relations of (1) giving wk0=wk0+1 in Mthen have the form bc(b) = ac(a). Since wk0 and bc(b) both express ∆ in M, we have wk0=bc(b), which contradicts the minimality assumption on m. Since all the letters of wjlie inside Aifor j∈[[1, m]], we have that the relations of (1) giving wj=wj+1 for j∈[[1, m −1]] also hold in Mi. Thus w1is a word in Ai, which expresses the element ∆iin Mi. Let now s∈Sbe a simple element of M. By definition, there is a word w1=ww2in A expressing ∆ such that wexpresses sin M. The first part of the proof gives that there is some i∈[[1, h]] such that w1, w and w2are actually words in Ai. The word w(resp. w2) then expresses an element es(resp. es′) in Mi, with eses′= ∆i. We then have es∈Siand ϕi(es) = s. Suppose now that s /∈ {1,∆}, and let w′be another word in Aexpressing sin M. By definition, there is a sequence of words w1,...,wnof words in A, such that w1=w, wn=w′and each wkis equivalent to wk+1 by the use of one relation of presentation (1). Since s≺∆, none of the wkcontain a subword expressing ∆ in M. We deduce that all UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 19 the relations giving wk=wk+1 in Mfor k∈[[1, n −1]] are between words in Aiand also hold in Mi. Thus wkalso expresses esfor all k∈[[1, n]].  Proposition 3.7. Let i∈[[1, n]] and let s∈Si. The morphism ϕiinduces an injective morphism of lattices from {t∈Si|ts}to {t∈S|tϕi(s)}. Furthermore, if s6= ∆i, then this morphism is also surjective. Proof. We show the injectivity and surjectivity assumptions before considering lcms and gcds. Let s∈Siand let Xand Ydenote the two considered sets. Since ϕiis a morphism of monoids, it restricts to a poset morphism X→Y. First, we show that ϕi:X→Yis always injective. Let t, t′∈Xbe such that ϕi(t) = ϕi(t′). Let wand w′be two words in Ai expressing tand t′, respectively. The words wand w′express the same element ϕi(t) in M. If ϕi(t) = ∆, then wand w′are two words in Airepresenting ∆ in M. The proof of Lemma 3.6 gives that wand w′represent the same element in Mi, which is ∆i. If ϕi(t)6= ∆, then wand w′express the same element in Siby Lemma 3.6, thus t=t′. Now, suppose that s6= ∆. We show that ϕi:X→Yis surjective. Let tϕi(s) in M. By definition, there is a word w=w1w2in Aexpressing ϕi(s) such that w1expresses t. By Lemma 3.6, wand w1are words in Ai. The word w1then expresses some element et∈Sisuch that ϕi(et) = tand ϕi:X→Yis surjective. We now show that ϕi:Si→Sis a morphism of lattices. That is, ϕipreserves rightlcms and left-gcds. Let s, t ∈Sibe two simples of (Mi,∆i), and let u=s∧t. Let x:= ϕi(s)∧ϕi(t). Since the simple ϕi(u) is obviously a left-divisor of ϕi(s) and ϕi(t), we have ϕi(u)x. If s= ∆i, then u=tand x=ϕi(t) = ϕi(u). Likewise the result is clear if t= ∆i. We assume from now on that t, s 6= ∆i. Since s6= ∆i(resp. t6= ∆i), the first part of the proof gives the existence of a unique ex(resp. ex′) in Sisuch that exs(resp. ex′t) and ϕi(ex) = x=ϕi(ex′). Since ϕiis injective on Si, we get that ex=ex′is a common divisor of sand tin Mi. We obtain exu,xϕi(u) and x=ϕi(u). Let now v=s∨tand y=ϕi(s)∨ϕi(t). Again, since ϕi(v) is an obvious right-multiple of both sand t, we have yϕi(v). By the first part of the proof, there is a unique ey∈Si such that eyvand ϕi(ey) = y. The first part of the proof also gives that s, t ey. Thus vey,ϕi(v)ϕi(ey) = yand ϕi(v) = y. Of course, one can show by similar arguments that ϕiis an injective morphism of lattices from (Si,) to (S, ). Corollary 3.8. Let s, t be two simple elements in M, both different from ∆and 1. Assume that s=ϕi(es)and t=ϕj(et)for i6=j,es∈Siand et∈Sj. We have s∧t= 1 and s∨t= ∆ in M. Furthermore, the word st is greedy in (M, ∆). Proof. Let u:= s∧t. By Proposition 3.7, there is a unique eu∈Si(resp. eu′∈Sj) such that ϕi(eu) = ϕj(eu′) = u. Since i6=j, Lemma 3.6 gives that u= 1. We apply similar reasoning for lcms. This gives in particular that s∧t= 1, thus the path st is greedy.  Proposition 3.9. Let i∈[[1, h]]. The morphism ϕi:G(Mi)→G(M)preserves leftweighted factorizations. In particular it is injective. Proof. We first show that ϕi:Mi→Mpreserves greedy normal forms. By definition, we only have to show that, if st is a greedy word in Mi, then ϕi(s)ϕi(t) is a greedy word in M. If st is a greedy word of length 2 in Mi, then Proposition 3.7 gives ϕi(s)∧ϕi(t) = ϕi(s)∧ϕi(t) = ϕi(s∧t) = 1. The word ϕi(s)ϕi(t) is then greedy by Lemma 1.3. Let now x= ∆k is1···srbe the left-weighted factorization of some x∈Mi. Since ∆i6 s1···sr, we have ∆ 6 ϕi(s1···sr). Furthermore, the word ϕi(s1)···ϕi(sr) is greedy because ϕi:Mi→Mpreserves greediness. The word ∆kϕi(s1)···ϕi(sr) is then the left-weighted factorization of ϕi(x) by definition.  20 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS Proposition 3.10. Let i∈[[1, h]], and let x∈G(Mi). We have ϕi(SSS(x)) = SSS(ϕi(x)). Furthermore, if xis not conjugate to a power of ∆iin Mi, then the centralizer of ϕi(x)in G(M)is the image under ϕiof the centralizer of xin G(Mi). Proof. Let xbe an element of G(Mi). By Proposition 3.9, we have ϕi(init(x)) = init(ϕi(x)) and ϕi(fin(x)) = fin(ϕi(x)). Thus, we also have ϕi(cyc(x)) = cyc(ϕi(x)) and ϕi(dec(x)) = dec(ϕi(x)). If xis conjugate to some ∆k iwith k∈Z, then we have SSS(x) = {∆k}= SSS(ϕi(x)). From now on, we suppose that xis not conjugate to any power of ∆. We show that inf(SSS(x)) = inf(SSS(ϕi(x)) and sup(SSS(x)) = sup(SSS(ϕi(x)). First, let x′∈SSS(x), Proposition 3.9 gives that inf(SSS(ϕi(x))) ⩾inf(ϕi(x′)) = inf(x′) = inf(SSS(x)), sup(SSS(ϕi(x))) ⩽sup(ϕi(x′)) = sup(x′) = sup(SSS(x)). Conversely, one can reach an element yof SSS(ϕi(x)) by applying a sequence of cycling and decycling to ϕi(x). Applying the same operations to xgives a conjugate eyof xin G(Mi) such that ϕi(ey) = y. We then have inf(SSS(x)) ⩾inf(ey) = inf(y) = inf(SSS(ϕi(x))), sup(SSS(x)) ⩽sup(ey) = sup(y) = sup(SSS(ϕi(y))). Thus, inf(SSS(x)) = inf(SSS(ϕi(x))) and sup(SSS(x)) = sup(SSS(ϕi(x))) as claimed. Now, we show that ϕi(SSS(x)) ⊂SSS(ϕi(x)). Let x′∈SSS(x), we have inf(ϕi(x′)) = inf(x′) = inf(SSS(x)) = inf(SSS(ϕi(x))), and likewise, sup(ϕi(x′)) = sup(SSS(ϕi(x))). Since ϕi(x′) is conjugate to ϕi(x), we have ϕi(x′)∈SSS(ϕi(x)). Let x′∈SSS(x), and let s∈Sbe a simple element of Msuch that ϕi(x′)s∈SSS(ϕi(x)). We show that s∈ϕi(Si). Let x′= ∆k is1···srbe the left-weighted factorization of x′in Mi. The left-weighted factorization of ϕi(x′) is given by ∆kϕi(s1)···ϕi(sr). We have ϕi(x′)s= ∆k−1φk−1(s)ϕi(s1)···ϕi(sr)s. If s /∈ϕi(Si), then the words φk−1(s)ϕi(s1) and ϕi(sr)sare in greedy normal form by Corollary 3.8. The expression above is then the left-weighted factorization of ϕi(x′)sin M. Thus inf(ϕi(x′)s) = k−1 and ϕi(x′)s/∈SSS(ϕi(x′)). The same reasoning shows that if ϕi(x′)s−1∈SSS(ϕi(x′)), then s∈ϕi(Si). An immediate induction then shows that the connected component of ϕi(x′) in CG(ϕi(x)) is contained in ϕi(SSS(x)). Since CG(ϕi(x)) is connected, we have SSS(ϕi(x)) ⊂ϕi(SSS(x)). This also shows that the conjugacy graph CG(ϕi(x)) is the image under of CG(x) under ϕi, whence the result on centralizers.  Proposition 3.11. (Periodic elements in a ∆-product) Let p, q be nonzero integers, and let ρ∈G(M)be a (p, q)-periodic element. There is some i∈[[1, h]] and some (p, q)-periodic element σ∈G(Mi)such that ρis conjugate to ϕi(σ)in G(M). Proof. Let ρ∈G(M) be a (p, q)-periodic element. If ρis conjugate to a power of ∆, then the result is obvious. Otherwise, Proposition 1.11 gives that ρis conjugate to an element of the form ∆ksfor some s∈S. By Proposition 3.5, there is some i∈[[1, h]] and some es∈S such that ϕi(es) = s. We then have ∆ks=ϕi(∆k ies). The element ∆k iesis (p, q)-periodic in Miby Proposition 3.9.  Let i∈[[1, h]] and let kibe the smallest integer such that ∆ki iis central in G(Mi). By Proposition 3.5, the smallest central power of ∆ in G(M) is given by the lcm of the kifor i∈[[1, h]]. UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 21 Proposition 3.12. (Center of a nontrivial ∆-product) Assume that h⩾2. The intersection in G(M)of all the ϕi(G(Mi)) for i∈[[1, h]] is the subgroup generated by ∆. The center of G(M)is cyclic and generated by the smallest central power of ∆in G(M). Proof. Since h⩾2, we can choose i6=jin [[1, h]]. Let xi∈G(Mi) and xj∈G(Mj) be such that x:= ϕi(xi) = ϕj(xj)∈G(M). If xi= ∆ki is1. . . sr(resp. xj= ∆kj jt1. . . tu) is the left-weighted factorization of xiin G(Mi) (resp. of xjin G(Mj)), then by Proposition 3.9, the left-weighted factorization of xin G(M) is then given by x= ∆kiϕi(s1)···ϕi(sr) = ∆kjϕj(t1)···ϕj(tu). We deduce that ki=kj,r=uand ϕi(sk) = ϕj(tk) for all k∈[[1, r]]. Since i6=j, Lemma 3.6 gives that r= 0 and xis a power of ∆. Let now xbe an element of Z(G(M)), and let i∈[[1, h]]. By definition, xlies in the centralizer of ϕi(a) for all a∈Ai. Since all the Miare distinct from M(1,1), the elements of Aiare not conjugate to a power of ∆. Thus Proposition 3.10 gives that xactually lies in the image under ϕiof the common centralizer of all elements of Aiin Mi. In other words we have x∈ϕi(Z(G(Mi)). We then have that x∈Th i=1 ϕi(Z(G(Mi))) is a power of ∆, whence the result.  3.2. Hosohedral-type Garside groups. Definition 3.13. A Garside monoid (M, ∆) is a hosohedral-type monoid if it is a ∆-product of circular monoids. The enveloping group of a hosohedral-type monoid is a hosohedral-type group. For instance, torus knot groups are hosohedral-type groups. Indeed, for p, q two positive coprime integers, we have ha, b |ap=bqi ≃ G(M(1, p)∗∆M(1, q)) In his PhD thesis, Picantin introduced hosohedral-type monoids as the Garside monoids whose lattice of simple elements satisfy a strong property regarding gcds and lcms. Theorem 3.14. [Pic00, Proposition 2.5] A finite lattice (S, ∧,∨,0,1) has hosohedral-type if any couple (s, t)in Ssatisfies (a∧b, a ∨b)∈ {(a, b),(b, a),(0,1)} A Garside monoid (M, ∆) is a hosohedral-type monoid if and only if the lattice (S, )of its simple elements has hosohedral-type. Note that this theorem also covers the case of non-homogeneous Garside monoids, which we do not consider here. In his proof, Picantin directly classifies all the Garside monoids whose lattice of simples is a fixed hosohedral-type lattice, in terms of the length of maximal chains. Under the notation of [Pic00, Proposition VI.2.5 and Definition VI.2.7], the monoid fus+[[hu1,1 1·...·hu1,k1 1·hu2,1 2·...·hun,kn n]] is isomorphic to the ∆-product M(u1,1, h1)∗∆· · · ∗∆M(u1,k1, h1)∗∆M(u2,1, h2)∗∆· · · ∗∆M(un,kn, hn). This property regarding lcms and gcds of simple elements also induces strong geometric properties for the associated Garside group, as pointed out by Mireille Soergel in [Soe23, Theorem 4.6]. By [DDGKM, Proposition VI.1.11], every Garside monoid (M, ∆) with set of simples Sadmits a presentation (2) M≃ hS|s·t=st ∀s, t ∈Ssuch that st ∈Si+ 22 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS Theorem 3.15. [Soe23, Theorem 4.6] Let (M, ∆) be a Garside monoid. The flag complex of the Cayley graph of G(M)associated to presentation (2) is systolic (in the sense of [Soe23, Section 2]) if and only if (M, ∆) is a hosohedral-type monoid. Again this theorem also covers the case of non-homogeneous Garside monoids. Example 3.16.In [Soe23, Question after Remark 4.8], Mireille Soergel considers the Garside group Gk, defined for an integer k⩾2 by the presentation Gk:= ha, b |aba =bki. She asks whether or not this group is isomorphic to a hosohedral-type group. We have the following isomorphisms of groups, given by Tietze transformations ha, b |aba =bki=ha, b, x |x=bka−1, aba =bki =ha, b, x |a=x−1bk, aba =bki =hb, x |x−1bkbx−1bk=bki =hb, x |bk+1 =x2i. The last group is a hosohedral-type group. More generally, a conjecture of Picantin ([Pic00, Conjecture 1]) states that, if Mis a Garside monoid with two atoms, then the enveloping group G(M) is isomorphic to either a group of the form ha, b |ap=bqifor positive integers p, q, or to an Artin group of dihedral type. In either case this would mean that the enveloping group of a Garside monoid with two generators is always a hosohedral-type group. From now on, let M=M(m1, ℓ1)∗∆···M(mh, ℓh) be a hosohedral-type monoid. Using Remark 3.3, we assume that (mj, ℓj)6= (1,1) for all j∈[[1, h]]. We also assume that h⩾2, otherwise we recover results from Section 2. By Proposition 3.9, we can identify the factors M(mj, ℓj) (resp. G(mi, ℓi)) with the associated subgroup of M(resp. of G(M)). To avoid confusion, the simple s(i, p) of the factor M(mj, ℓj) will be denoted by sj(i, p). 3.2.1. Conjugacy. Like in the case of circular groups, an element in a super-summit set of a hosohedral-type monoid is either rigid or periodic. Proposition 3.17. Let xbe an element of G(M). If xlies in its own super-summit set, then it is either rigid or periodic. Proof. The proof imitates the case of circular groups. First, if inf(x) = sup(x), then we have x= ∆kfor some k∈Z. In this case, xis obviously both rigid and (1, k)-periodic. Suppose now that sup(x) = inf(x) + 1. We have x= ∆ksj(i, p) for some j∈[[1, h]], i∈[[0, mj−1]] and 0 < p < ℓj. The element xis periodic if and only if it is periodic as an element of the factor G(mj, ℓj). By Lemma 2.8, this is equivalent to p+kℓj≡0[mj]. If this is not the case, the word sj(i, p)φ−k(sj(i, p)) is greedy in M(mi, ℓi). It is then also greedy in Mby Proposition 3.9. Lastly, suppose that sup(x)>inf(x) + 1. The left-weighted factorization of xis given by ∆ks1···srwith r > 1. We claim that xis rigid. Otherwise, the word srφ−k(s1) is not greedy. By Corollary 3.8, this implies that srand s1lie in the same factor M(mi, ℓi). We can then apply the last part of the proof of Proposition 2.7 to show that we either have sup(cyc(x)) <sup(x) or inf(cyc(x)) >inf(x). In both cases, we have x /∈SSS(x).  By Proposition 3.11, there are (p, q)-periodic elements in G(M) if and only if there are (p, q)-periodic elements in some factor G(mj, ℓj). However, two (p, q)-periodic elements coming from two different factors are not conjugate in general. Proposition 3.18. Let p, q be integers. Two (p, q)-periodic elements of G(M)are conjugate if and only if they both admit a conjugate lying the the same factor G(mj, ℓj). UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 23 Proof. Let ρ, σ be two (p, q)-periodic elements in G(M). If ρand σare conjugate, then SSS(ρ) = SSS(σ). Let x∈SSS(ρ). We have either x= ∆ksfor some simple sand some integer kor x= ∆k. In both cases, xis a conjugate of both ρand σlying in a fixed factor M(mi, ℓi) by Proposition 3.5. Conversely, if xand yare respective conjugate of ρand σin M(mi, ℓi), then xand y are (p, q)-periodic elements of M(mi, ℓi). They are conjugate in G(mi, ℓi) (in particular in G(M)) by Proposition 2.12.  Example 3.19.Let nbe a positive integer, and consider M=M(1, n)∗∆M(1, n) = ha, b |an=bni+. By Proposition 3.10, we have SSS(a) = {a}and SSS(b) = {b}, thus a and bare two (n, 1)-periodic elements that are not conjugate in G(M). On the other hand, the conjugacy of non-periodic elements in a hosohedral-type group behaves in the same way as in a circular group. Proposition 3.20. Let x∈G(M)be a non-periodic element. The super-summit set of x is made of rigid elements. The only arrows starting from an object yof CG(x)are labeled by init(y)and fin(y). In particular, one can go from an element of SSS(x)to any other by a finite sequence of cycling, decycling and application of the Garside automorphism. Proof. We mimic the proof of Proposition 2.14. Let y∈SSS(x). Since xis not periodic, yis not periodic. It is then rigid by Proposition 3.17, and we have sup(y)>inf(y). We assume that the left-weighted factorization of yis ∆ksj1(i1, p1)···sjr(ir, pr). Let sbe a simple element of M. We have ys= ∆k−1φk−1(s)sj1(i1, p1)···sjr(ir, pr)s. If s∈ {1,∆}, then ys∈SSS(x) is obvious. Otherwise, let jbe such that slies in the factor M(mj, ℓj). If j /∈ {j1, jr}, we have inf(ys) = k−1 and y /∈SSS(x). - If jr=j1, then the proof of Proposition 2.14 gives that ys∈SSS(x) if and only if s∈ {init(y),fin(y)}. - If jr6=j1and j=j1, then the word sjr(ir, pr)sis greedy in Mby Corollary 3.8. We then have ys∈SSS(x) if and only if φk−1(s)sj1(i1, pr) = ∆, i.e. if s= init(y). - If jr6=j1and j=jr, then the word φk−1(s)sj1(i1, p1) is greedy in Mby Corollary 3.8. We then have ys∈SSS(x) if and only if sjr(ir, pr)s= ∆, i.e. if s=fin(y).  Theorem 3.21. Let Mbe a hosohedral-type monoid. If α, β ∈G(M)are such that αn=βn for some nonzero integer n, then we either have -αand βare conjugate. -αand βare nonconjugate periodic elements of G(M). Proof. Again, αis periodic if and only if its power αnis periodic, if and only if βis periodic. Suppose that αand βare not nonconjugate periodic elements of G(M). Then αand β are either conjugate periodic elements (in which case they are conjugate) or non-periodic elements. Up to replacing α, β with α−1, β−1, we now assume that n > 0. We assume from now on that aand bare non-periodic elements of G(M). Up to conjugacy, we can assume that a∈SSS(a). By Proposition 3.17, we have that ais rigid. The element xis then rigid as a power of the rigid element a. Let now c∈G(M) be so that bc∈SSS(b). Since bis not periodic, bcis rigid as well as xc= (bc)n. We have x, xc∈SSS(x). By Proposition 3.20, there is an finite sequence of cycling, decycling, and applications of the Garside automorphism sending xto xc. By Lemma 1.9, applying the same transformations to agives a rigid element a′whose n-th power is xc. Again by Lemma 1.9, we have a′=bcand thus aand bare conjugate.  24 UNIQUENESS OF ROOTS IN CIRCULAR AND HOSOHEDRAL-TYPE GARSIDE GROUPS 3.2.2. Partial computations of homology. Let M=M(m1, ℓ1)∗∆· · · ∗∆M(mh, ℓh) be a hosohedral-type monoid. We can construct the Dehornoy-Lafont complex (cf. Section 2.3.1) associated to Mto try and compute the homology of G(M). Since the lcm of two distinct atoms of a hosohedral-type group is ∆ by Corollary 3.8 and Lemma 2.4, we can mimic the proof of Lemma 2.17 to get Lemma 3.22. If Mis a hosohedral-type monoid with matoms, we order its atoms by a0< a1< . . . < am−1. We have X0={[∅]},X1={[a0],...,[am−1]},X2={[a0, ai]|i∈ [[1, m −1]]}and Xn=∅for n⩾3. Like in the case of a circular monoid, this implies that a hosohedral-type group has homological dimension at most 2. Furthermore, as H2(G(M),Z) is free, the computation of G(M)ab is sufficient to determine all the integral homology of G(M). Unfortunately, we are only able to compute the free part of G(M)ab. Proposition 3.23. Let M=M(m1, ℓ1)∗∆· · · ∗∆M(mh, ℓh)be a hosohedral-type monoid. For i∈[[1, h]], denote by dithe gcd of mi∧ℓi. The free part of G(M)ab has rank 1 + Ph i=1(di−1). Proof. First, let j∈[[1, h]]. Let also e0,...,ed−1denote the canonical basis of Zdj. By Lemma 2.18, the morphism sj(i, 1) 7→ ei′, where i′is the remainder in the Euclidean division of iby djinduces an isomorphism between G(mj, ℓj) and G(m, ℓ)ab ≃Zdj. Under this isomorphism, we have ∆j=sj(0, m)7→   dj−1 X i=0 ei  mj dj = dj−1 X i=0 mj dj ei. Let A≃ZPn i=1 dibe the direct product of the G(mi, ℓi)ab for i∈[[1, h]]. The isomorphisms G(mj, ℓj)ab ≃Zdjdescribed above induce a morphism pfrom the free product of the G(mi, ℓi) to A, and we have A= (G(m1, ℓ1)∗ · · · ∗ G(mh, ℓh))ab. We then have G(M)ab =A.hp(s1(0, m1)) −p(si(0, mi)),∀i∈[[2, h]]i as the vectors p(s1(0, m1)) −p(si(0, mi)) are linearly independent, they span a submodule of Aof rank h−1. The free part of G(M)ab then has rank h X i=1 di−h+ 1 = 1 + h X i=1 (di−1).  Example 3.24.Let p, q ⩾2 be integers, and consider the group G:= ha, b |ap=bqi ≃ G(M(1, p)∗∆M(1, q)). 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