Journal Article: The Bieri-Neumann-Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group
Koban, Nic and Piggott, Adam (2014) The Bieri-Neumann-Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group. Illinois Journal of Mathematics, 58 1: 27-41.
Journal Article: On the automorphisms of a graph product of abelian groups
Gutierrez, Mauricio, Piggott, Adam and Ruane, Kim (2012) On the automorphisms of a graph product of abelian groups. Groups, Geometry, and Dynamics, 6 1: 125-153. doi:10.4171/GGD/153
Journal Article: The automorphism group of the free group of rank 2 is a CAT(0) group
Piggott, Adam, Ruane, Kim and Walsh, Genevieve S. (2010) The automorphism group of the free group of rank 2 is a CAT(0) group. Michigan Mathematical Journal, 59 2: 297-302. doi:10.1307/mmj/1281531457
Journal Article: Palindromic primitives and palindromic bases in the free group of rank two
Piggott, Adam (2006) Palindromic primitives and palindromic bases in the free group of rank two. Journal of Algebra, 304 1: 359-366. doi:10.1016/j.jalgebra.2005.12.005
Combinatorial and geometric group theory
The symmetries of an object have an algebraic structure. Group theory is the study of such structures. In combinatorial group theory, groups are specified via group presentations. This means that we specify an alphabet of symbols, often only a few symbols, and some algebra rules which hold in the group. Everything else about the group must be deduced from the rules we specify. In geometric group theory, we exploit deep connections between groups and geometric structures. There is a sense in which a group itself is a geometric object, and every geometric object comes equipped with a group of symmetries (the isometries). We can use geometry to learn about groups, and we can learn about geometric structures using group theory.
I will be happy to talk to any honours, masters or Ph.D. student interested in combinatorial and/or geometric group theory to see if we can find a topic which suits their interests. An interested student may wish to peruse the book Office Hours with a Geometric Group Theorist, edited by Matt Clay & Dan Margalit to get a feel for some of the topics in geometric group theory. For combinatorial group theory I suggest browsing through Combinatorial Group Theory by Lyndon and Shupp.
Koban, Nic and Piggott, Adam (2014) The Bieri-Neumann-Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group. Illinois Journal of Mathematics, 58 1: 27-41.
On the automorphisms of a graph product of abelian groups
Gutierrez, Mauricio, Piggott, Adam and Ruane, Kim (2012) On the automorphisms of a graph product of abelian groups. Groups, Geometry, and Dynamics, 6 1: 125-153. doi:10.4171/GGD/153
The automorphism group of the free group of rank 2 is a CAT(0) group
Piggott, Adam, Ruane, Kim and Walsh, Genevieve S. (2010) The automorphism group of the free group of rank 2 is a CAT(0) group. Michigan Mathematical Journal, 59 2: 297-302. doi:10.1307/mmj/1281531457
Palindromic primitives and palindromic bases in the free group of rank two
Piggott, Adam (2006) Palindromic primitives and palindromic bases in the free group of rank two. Journal of Algebra, 304 1: 359-366. doi:10.1016/j.jalgebra.2005.12.005
Piggott, Adam (2018). Coxeter groups. In Dan Margalit and Matt Clay (Ed.), Office hours with a geometric group theorist () Princeton, NJ, United States: Princeton University Press. doi:10.23943/princeton/9780691158662.003.0013
Recognizing right-angled Coxeter groups using involutions
Cunningham, Charles, Eisenberg, Andy, Piggott, Adam and Ruane, Kim (2016) Recognizing right-angled Coxeter groups using involutions. Pacific Journal of Mathematics, 284 1: 41-77. doi:10.2140/pjm.2016.284.41
CAT(0) extensions of right-angled Coxeter groups
Cunningham, Charles, Eisenberg, Andy, Piggott, Adam and Ruane, Kim (2016) CAT(0) extensions of right-angled Coxeter groups. Topology Proceedings, 48 277-287.
On groups presented by monadic rewriting systems with generators of finite order
Piggott, Adam (2015) On groups presented by monadic rewriting systems with generators of finite order. Bulletin of the Australian Mathematical Society, 91 3: 426-434. doi:10.1017/S0004972715000015
Koban, Nic and Piggott, Adam (2014) The Bieri-Neumann-Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group. Illinois Journal of Mathematics, 58 1: 27-41.
The symmetries of McCullough-Miller space
Piggott, Adam (2012) The symmetries of McCullough-Miller space. Algebra and Discrete Mathematics, 14 2: 239-266.
On the derived length of coxeter groups
Brooksbank, Peter A. and Piggott, Adam (2012) On the derived length of coxeter groups. Communications in Algebra, 40 3: 1142-1150. doi:10.1080/00927872.2010.547539
On the automorphisms of a graph product of abelian groups
Gutierrez, Mauricio, Piggott, Adam and Ruane, Kim (2012) On the automorphisms of a graph product of abelian groups. Groups, Geometry, and Dynamics, 6 1: 125-153. doi:10.4171/GGD/153
Piggott, Adam and Ruane, Kim (2010) Normal forms for automorphisms of universal coxeter groups and palindromic automorphisms of free groups. International Journal of Algebra and Computation, 20 8: 1063-1086. doi:10.1142/S0218196710006035
The automorphism group of the free group of rank 2 is a CAT(0) group
Piggott, Adam, Ruane, Kim and Walsh, Genevieve S. (2010) The automorphism group of the free group of rank 2 is a CAT(0) group. Michigan Mathematical Journal, 59 2: 297-302. doi:10.1307/mmj/1281531457
Rigidity of graph products of abelian groups
Gutierrez, Mauricio and Piggott, Adam (2008) Rigidity of graph products of abelian groups. Bulletin of the Australian Mathematical Society, 77 2: 187-196. doi:10.1017/S0004972708000105
Andrews–Curtis groups and the Andrews–Curtis conjecture
Piggott, Adam (2007) Andrews–Curtis groups and the Andrews–Curtis conjecture. Journal of Group Theory, 10 3: 373-387. doi:10.1515/jgt.2007.029
The manifestation of group ends in the Todd-Coxeter coset enumeration procedure
Piggott, Adam (2007) The manifestation of group ends in the Todd-Coxeter coset enumeration procedure. International Journal of Algebra and Computation, 17 1: 203-220. doi:10.1142/S0218196707003561
Palindromic primitives and palindromic bases in the free group of rank two
Piggott, Adam (2006) Palindromic primitives and palindromic bases in the free group of rank two. Journal of Algebra, 304 1: 359-366. doi:10.1016/j.jalgebra.2005.12.005
Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.
Combinatorial and geometric group theory
The symmetries of an object have an algebraic structure. Group theory is the study of such structures. In combinatorial group theory, groups are specified via group presentations. This means that we specify an alphabet of symbols, often only a few symbols, and some algebra rules which hold in the group. Everything else about the group must be deduced from the rules we specify. In geometric group theory, we exploit deep connections between groups and geometric structures. There is a sense in which a group itself is a geometric object, and every geometric object comes equipped with a group of symmetries (the isometries). We can use geometry to learn about groups, and we can learn about geometric structures using group theory.
I will be happy to talk to any honours, masters or Ph.D. student interested in combinatorial and/or geometric group theory to see if we can find a topic which suits their interests. An interested student may wish to peruse the book Office Hours with a Geometric Group Theorist, edited by Matt Clay & Dan Margalit to get a feel for some of the topics in geometric group theory. For combinatorial group theory I suggest browsing through Combinatorial Group Theory by Lyndon and Shupp.