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Residual Finiteness of Certain 2-Dimensional Artin Groups

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Residual Finiteness of Certain 2-Dimensional Artin Groups RESIDUAL FINITENESS OF CERTAIN 2-DIMENSIONAL ARTIN GROUPS KASIA JANKIEWICZ Abstract. We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels ≥ 4. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed. A group G is residually finite if for every g 2 G − f1g there exists a finite quotient φ : G ! G¯ such that φ(g) 6= 1. The main goal of this paper is to extend this list of Artin groups known to be residually finite. Let Γ be a simple graph where each pair of vertices a; b in Γ is labelled by an integer Mab ≥ 2. The associated Artin group ArtΓ = ha 2 V (Γ) j (a; b)Mab = (b; a)Mab for a; b joined by an edgei: By (a; b)Mab we denote the alternating word abab : : : of length Mab. The Artin group on two generators with the label M will be denoted by ArtM , and the Artin group with three generators and labels M; N; P will be denoted by ArtMNP . Theorem A. If M; N; P ≥ 4, then the Artin group AMNP is residually finite. None of the groups in Theorem A with M; N; P < 1 were previously known to be residually finite.We also obtain residual finiteness of many more 2-dimensional Artin groups. For precise statements see Section 6. In a subsequent work [Jan21] we prove the residual finiteness of Artin groups Art2MN where M; N ≥ 4 and at least one of them is even. Our proof of Theorem A relies on a splitting of these Artin group as a free product with amal- gamation or HNN extension of finite rank free groups. The existence of such splitting depends on the combinatorics of the defining graph. Recall an Artin group ArtΓ with the defining graph Γ has large type if all labels in Γ are at least 3. The quotient of an Artin group, obtained by adding the 2 relation a = 1 for every a 2 V (Γ) is a Coxeter group. We say ArtΓ is spherical if the corresponding Coxeter quotient is finite, and ArtΓ is 2-dimensional if no triple of generators generates a spherical Artin group. In particular, every large type Artin group is 2-dimensional. For the definition of admissible partial orientation of Γ, see Definition 4.2. We prove the following. Theorem B. If Γ admits an admissible partial orientation, then ArtΓ splits as a free product with amalgamation or an HNN-extension of finite rank free groups. The above theorem includes all large type Artin groups whose defining graph Γ admits an orientation such that each cycle is directed. All linear groups are residually finite by a classical result by Mal'cev [Mal40]. Among Artin group very few classes are known to be residually finite, and even fewer linear. It was once a major open question whether braid groups are linear and it was proved independently by Krammer [Kra02] and Bigelow [Big01]. Later, the linearity was extended to all spherical Artin groups by Cohen- Wales [CW02], and independently by Digne [Dig03]. The right-angled Artin groups are also well known to be linear. Since linearity is inherited by subgroups, any virtually special Artin group 2010 Mathematics Subject Classification. 20F36, 20F65, 20E26. Key words and phrases. Artin groups, Residual finiteness, Graphs of free groups. 1 is linear. Artin groups whose defining graphs are forests are the fundamental groups of graph manifolds with boundary by the work of Brunner [Bru92] and Hermiller-Meier [HM99], and so they are virtually special by the work of Liu [Liu13] and Przytycki-Wise [PW14]. Artin groups in certain classes (including 2-dimensional, 3-generators) are not cocompactly cubulated even virtually, unless they are sufficiently similar to RAAGs by Huang-Jankiewicz-Przytycki [HJP16] and independently by Haettel [Hae20b]. In particular, if M; N; P are finite, none of the groups in Theorem A, is virtually cocompactly cubulated. Haettel has a conjectural classification of all virtually cocompactly cubulated Artin groups [Hae20b]. Haettel also showed that some triangle-free Artin group act properly but not cocompactly on locally finite, finite dimensional CAT(0) cube complexes [Hae20a]. We note that if one of the exponents M; N; P is infinite, then the residual finiteness of ArtMNP is well-known. The list of other known families of residually finite Artin groups is short. An Artin group is of FC type if every clique (i.e. a complete induced subgraph) in Γ is the defining graph of a spherical Artin group. (Blasco-Garcia)-(Martinez-Perez)-Paris showed that FC type Artin groups with all labels even are residually finite [BGMPP19]. (Blasco-Garcia)-Juhasz-Paris showed in [BGJP18] the residual finiteness of Artin groups with defining graph Γ where the vertices of Γ admit a partition P such that • for each X 2 P the Artin group AX is residually finite, • for each distinct X; Y 2 P there is at most one edge in Γ joining a vertex of X with a vertex of Y , and • the graph Γ=P is either a forest, or a triangle free graph with even labels. The graph Γ=P is defined as follows. The vertices of Γ=P are P, and an edge with label M joins sets X; Y 2 P if there exist a 2 X; b 2 Y such that Mab = M. In [Jan21] the author proves the residual finiteness of Artin groups Art2MN where M; N ≥ 4 and at least one of them is even. The residual finiteness of 3-generator affine Artin groups (i.e. corresponding to affine Coxeter groups), i.e. Art244, Art236, Art333 follows from the work of Squier [Squ87]. Squier proved that Art244 splits as an HNN extension of F2 by an automorphism of an index two subgroup, and both Art236 and Art333 split as F3 ∗F7 F4 where F7 is normal and of finite index in each of the factors. We give a geometric proof of the Squier's splitting of Art333 in Example 4.14. The subgroup F7 has index three and two respectively in the factors F3 and F4 in the splitting of Art333. This yields a short exact sequence of groups 1 ! F7 ! Art333 ! Z=3 ∗ Z=2 ! 1: In particular Art333 is free-by-(virtually free), and therefore virtually free-by-free. Since every split extension of a finitely generated residually finite group by a residually finite group is residually finite [Mal56], we can conclude that Art333 is residually finite. Similar arguments yield residual finiteness of Art244 and Art236. The residual finiteness of Art333 and Art244 also follows from the fact that they are commensurable with the quotients of spherical Artin groups modulo their centers, respectively Art233 =Z and Art234 =Z [CC05]. Theorem B provides a splitting of ArtΓ. In general, the existence of a splitting does not guarantee residual finiteness. In order to prove Theorem A we carefully analyze the splitting and use a criterion for residual finiteness of certain amalgams of special form. See Theorem 2.8 and Theorem 2.11. The following question is open in general. Question. Let A; B; C be finite rank free groups. When is the group A ∗C B (or A∗B) residually finite? One instance where G = A∗C B (or A∗B) is residually finite is when C is malnormal in A; B. By the combination theorem of Bestvina-Feighn [BF92], if A; B are hyperbolic, and C is quasi-convex 2 in both A and B and malnormal in at least one of A; B, then G = A ∗C B is hyperbolic. Wise showed that in such a case, G is residually finite [Wis02], and later Hsu-Wise proved that G is in fact virtually special [HW15]. Another class of examples of residually finite amalgams are doubles of free groups along a finite index subgroup. These groups are virtually direct products of two finite rank free groups [BDGM01]. On the other hand there are examples of amalgamated products of free groups that are not residually finite. First such an example, which is an amalgam of two free groups along a common subgroup of finite index in each of the factors, was constructed by Bhattacharjee [Bha94]. More examples are lattices in the automorphism group of a product of two trees, which split as twisted doubles of free groups along a finite index subgroup, and they were constructed by [Wis96] and [BM97]. The Burger-Mozes examples are not only non residually finite, but virtually simple. The paper is organized as follows. In Section 1 we fix notation and recall some geometric group theory tools that we use later. In Section 2 we recall some facts about residual finiteness and prove our criterion for residual finiteness of twisted doubles of free group (Theorem 2.8) and of HNN extensions of free groups (Theorem 2.11). In Section 3 we recall the definition of Artin groups, and describe their non-standard presentations due to Brady-McCammond [BM00]. In Section 4 we carefully study the presentation complex from the previous section and prove Theorem B (as Theorem 4.3). Finally, in Section 5 we prove Theorem A (as Corollary 5.7 and Corollary 5.12). A proof in the case where at least one label is even, is generalized to a broader family of Artin groups in Section 6.
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