SPLITTINGS OF THREE GENERATOR ARTIN GROUPS

We show that a three generator Artin group AMNP where M ≤ N ≤ P splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either M > 2, or N > 3. We also prove that all even three generator Artin groups are residually finite. An Artin group on three generators is given by the presentation AMNP = 〈a, b, c | (a, b)M = (b, a)M , (b, c)N = (c, b)N , (c, a)P = (a, c)P 〉, where (a, b)M denote the alternating word aba . of length M , when M is finite, and the empty word if M = ∞. Squier showed that the 3-generator Euclidean Artin group, i.e. A236, A244 and A333, split as amalgamated products or an HNN extension of finite rank free groups along finite index subgroups [Squ87]. We generalize that result to other 3-generator Artin groups. Theorem A. Suppose that M ≤ N ≤ P where either M > 2, or N > 3. Then the Artin group AMNP splits as an amalgamated product or an HNN extension of finite rank free groups. The assumption of the above theorem are satisfied for all triples of numbers except for (2, 2, P ) and (2, 3, P ). An Artin group is spherical, if the associated Coxeter group is finite. A three generator Artin group AMNP is spherical exactly when 1 M + 1 N + 1 P > 1, i.e. (M,N,P ) = (2, 2, P ) or (2, 3, 3), (2, 3, 4), (2, 3, 5). All spherical Artin groups have infinite center, and none of them splits as a graph of finite rank free groups (see Proposition 2.10). The remaining cases are (2, 3, P ) where P ≥ 6 or p = ∞. The above theorem holds for triple (2, 3, 6) due to Squier [Squ87], and for (2, 3,∞). It remains unknown for (2, 3, P ) with P ≥ 7. The cases where M > 2 were considered in [Jan20, Thm B] and it was proven that they all split as amalgamated products of finite rank free groups. Graphs of free groups form an important family of examples in geometric group theory. Graph of free groups with cyclic edge groups that contain no Baumslag-Solitar subgroups are virtually special [HW10], and contain quasiconvex surface subgroups [Wil18]. Graphs of free groups with arbitrary edge groups can exhibit various behaviors. For example, an amalgamated product A ∗C B of finite rank free groups where C is malnormal in A,B is hyperbolic [BF92], and virtually special [HW15]. On the other hand there are examples of amalgamated products of finite rank free groups that are not residually finite [Bha94], [Wis96], and even simple [BM97]. The last two arise as lattices in the automorphism group of a product of two trees. By further analysis of the splitting, we are also able to show that some of the considered Artin groups are residually finite. Theorem B. The Artin group A2MN where M,N ≥ 4 and at least one of M,N is even, is residually finite. 1 An Artin group AMNP is even if all M,N,P are even. The above theorem combined with our result in [Jan20] (and the fact that A22P = Z× AP is linear) gives us the following. Corollary C. All even Artin groups on three generators are residually finite. All linear groups are residually finite [Mal40], so residual finiteness can be viewed as testing for linearity. Spherical Artin groups are known to be linear ([Kra02],[Big01] for braid groups, and [CW02], [Dig03] for general spherical Artin groups). The right-angled Artin groups are also well known to be linear, but not much more is known about linearity of Artin groups. In last years, a successful approach in proving that groups are linear is by showing that they are virtually special. Artin groups whose defining graphs are forests are the fundamental groups of graph manifolds with boundary [Bru92],[HM99], and so they are virtually special[Liu13], [PW14]. Many Artin groups in certain classes (including 2-dimensional, or three generator) are not cocompactly cubulated even virtually, unless they are sufficiently similar to RAAGs [HJP16],[Hae20b]. In particular, the only (virtually) cocompactly cubulated three generator Artin group are A22M = Z×AM , AMN∞ where M,N are both even, and AM∞∞ = Z ∗AM . Some triangle-free Artin groups act properly but not cocompactly on locally finite, finite dimensional CAT(0) cube complexes [Hae20a]. In [Jan20] we showed that AMNP are residually finite when M,N,P ≥ 3, except for the cases where (M,N,P ) = (3, 3, 2p + 1) with p ≥ 2. Few more families of Artin groups are known to be residually finite, e.g. even FC type Artin groups [BGMPP19], and certain triangle-free Artin groups [BGJP18]. Organization. In Section 1 we provide some background. In Section 2 we prove Theorem A as Proposition 2.5 and Corollary 2.9. We also show that the only irreducible spherical Artin groups splitting as graph of finite rank free groups are dihedral. In Section 3 we recall a criterion for residual finiteness of amalgamated products and HNN extensions of free groups from [Jan20] and prove Theorem B.