Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups

By J. Crisp, Bert Wiest. 2003

130 citations recorded by the source corpus.

We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to right- angled Artin groups a result of Lyndon for free groups, we show that the Euler characteristic −1 surface group (given by the relation x 2 y 2 = z 2 ) never embeds in a right-angled Artin group. AMS Classification 20F36, 05C25; 05C25 Keywords Cubed complex, graph braid group, graph group, right-angled Artin group, configuration space

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/997132435ce857416c9c48fe077d23a89f48dfe0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Christopher J. Leininger
2. Matt Clay
3. Johanna Mangahas
4. Ashot Minasyan
5. Thomas Koberda
6. Ruth Charney
7. Jason A. Behrstock
8. Lucas Sabalka

In Geometry and Topology

1. Johanna Mangahas
2. Anthony Genevois
3. Christopher J. Leininger
4. Matt Clay
5. Thomas Koberda
6. Ruth Charney
7. Jason A. Behrstock
8. Sang-hyun Kim

In Mathematical Physics

1. Johanna Mangahas
2. Christopher J. Leininger
3. Matt Clay
4. Thomas Koberda
5. Anthony Genevois
6. Jason A. Behrstock
7. Ruth Charney
8. Sang-hyun Kim

Citations in the corpus

• An introduction to right-angled Artin groups (2006, unknown): “Crisp and Wiest [34], extending work of Droms, Servatius, and Servatius [48], show that with only 3 exceptions, any surface group embeds in a right-angled Artin group.”
• Embedability between right-angled Artin groups (2011, background): “Actually, the fundamental group of any closed surface with Euler characteristic less than −1 embeds into some right-angled Artin group [10].”
• Discrete Morse theory and graph braid groups (2004, result): “There are positive results: Crisp and Wiest [ 10 ] have recently shown that every graph braid group embeds in a right-angled Artin group.”
• The geometry of right angled Artin subgroups of mapping class groups (2010, background): “As noted in [11], each of the associated generators si1 , .”
• Hereditary conjugacy separability of right angled Artin groups and its applications (2009, background): “Abrams in [1]) – see [15], and some hyperbolic 3-manifold groups – see [13].”
• Partially commutative metabelian groups: centralizers and elementary equivalence (2009, background): “2. Fundamental groups of almost all surfaces are subgroups of partially commutative groups [ 2 ].”
• Divergence and quasimorphisms of right-angled Artin groups (2010, methodology): “ element which acts as a rank-one isometry of XΓ. In this case, QHg(G) is infinite dimensional. Right-angled Artin groups have been shown to have an extremely rich family of subgroups, cf. [BB], [HW], [CW]. In contrast, using Theorem 5.2 and BurgerMonod’s result on nonexistence of quasimorphisms on h”
• Tits alternatives for graph products (2011, background): “The family of subgroups of right angled Artin groups is very rich and includes many examples (see [4, 16, 15, 9, 25]).”