Embeddings of right-angled Artin groups into higher-dimensional Thompson groups

Abstract

In this paper, we construct embeddings of right-angled Artin groups into higher dimensional Thompson groups. In particular, we embed every right-angled Artin groups into n-dimensional Thompson groups, where n is the number of complementary edges in the defining graph. It follows that Z n * Z embeds into nV for every n ≥ 1. IntoductionThe Thompson group V is an infinite simple finitely presented group, which is described as a subgroup of the homeomorphism group of the Cantor set C. Brin [1] defined higher dimensional Thompson groups as generalizations of the Thompson group V = 1V . By definition, n-dimensional Thompson group n 1 V embeds into n 2 V when n 1 ≤ n 2 . Brin [1] showed that V and 2V are not isomorphic. Bleak and Lanoue [3] showed n 1 V and n 2 V are isomorphic if and only if n 1 = n 2 .In [4], Bleak and Salazar-Díaz proved that Z 2 * Z does not embed in V . Recently, Corwin and Haymaker [6] determined which right-angled Artin groups embed into V . Using the nonembedding result of [4], they showed that Z 2 * Z is the only obstruction for a right-angled Artin group to be embedded into V . On the other hand, Belk, Bleak and Matucci [2] proved that a rightangled Artin group embeds in nV with sufficiently large n. They took n to be the sum of the number of vertices and the number of complementary edges in the defining graph. They conjectured that a right-angled Artin group embeds into (n − 1)V if and only if the right-angled Artin group does not contain Z n * Z . Corwin [5] constructed embeddings of Z n * Z into nV for every n ≥ 2. It follows that every nV with n ≥ 2 does not embed into V .In this paper, we give another construction of embeddings of right-angled Artin groups into higher-dimensional Thompson groups. In particular, we may embed a right-angled Artin group into nV , where n is the number of 1

Embedding right-angled Artin groups into Brin–Thompson groups

“…the set of all pairs of generators that do not commute. Kato has subsequently strengthened this result to n = |E c | [11]. Kato's bound is not sharp, and it remains an open problem to determine the smallest n for which a given right-angled Artin group embeds into nV .…”

Embeddings into Thompson's groups from quasi-median geometry

“…Consequently, Theorem 3.29 applies, so that V κ splits as a semidirect product Γ D b (Q, x) where Γ is a graph product of free groups, so a right-angled Artin group, and [Bri04]. But nV does not embed into V , for instance because nV contains the free product Z 2 * Z (initially proved in [Cor13], and reproved in [BBM16] for n ≥ 5 and in [Kat18] for n ≥ 2), which is not a subgroup of V according to [BSD13, Theorem 1.5].…”

“…Proposition 5.4 implies that, if nV embeds into a braided diagram group, then it must embed into Thompson's group V since nV is simple according to [Bri04]. But nV does not embed into V , for instance because nV contains the free product Z 2 * Z (initially proved in [Cor13], and reproved in [BBM16] for n ≥ 5 and in [Kat18] for n ≥ 2), which is not a subgroup of V according to [BSD13, Theorem 1.5]. Now, we will deduce from Theorem 5.1 some criteria to show that a diagram group embeds into some Thompson's group and we will apply them to the examples we mentioned in Section 2.4.…”

Graph and wreath products in topological full groups of full shifts

“…In light of this, it was conjectured in [2] that Z d * Z is the correct obstruction for embeddability in the Brin-Thompson group (d − 1)V defined in [3]. In [2], it was shown that at least RAAGs with m nodes and n non-commuting relations embed in (m + n)V. In [6], it was proved that if G is a RAAG with Z d * Z ≤ G, then indeed G ≤ (d − 1)V, as predicted by the conjecture. In this paper, we refute the conjecture from [2] and completely settle the issue of RAAG embeddability: all (countable) RAAGs embed in nV for all n ≥ 2.…”