# Classifying spaces for proper actions and almost-flat manifolds

Lead Research Organisation:
University of Southampton

Department Name: Sch of Mathematical Sciences

### Abstract

In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives.

In the last twenty years, non-positively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of right-angled Artin groups, and to investigate Brown's conjecture.

Our second objective is on almost-flat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.

In the last twenty years, non-positively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of right-angled Artin groups, and to investigate Brown's conjecture.

Our second objective is on almost-flat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.

### Planned Impact

The proposed research lies in the field of Pure Mathematics and therefore it will mostly have academic impact. However, because of its proximity to areas such as Applied Combinatorial Group Theory and Applied Algebraic Topology, Crystallography, Physics and String Theory, there is a long-term impact on more applied scientific fields.

There are rapidly growing applications of non-positively curved groups and spaces in Biology, Robotics, and Computer Science. On the other hand, (almost-)crystallographic groups, (almost-)flat manifolds and orbifolds are fundamental objects in Crystallography, Particle Physics, and String Theory. In the proposed research we will address some fundamental questions regarding these groups and spaces such as constructing computationally good models for classifying spaces of groups for proper actions and investigating the existence of Spin structures on almost-flat manifolds. From the first part, there are foreseeable applications to other more applied branches of Mathematics via the Isomorphism Conjectures. In the second, we study the existence of Spin structures on almost-flat manifolds. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. Therefore, there are potential benefits of the proposed research to physicists, biologists, and computer scientists.

As most of the envisioned impact of the grant proposal will be long-term in nature, we will focus on the initial societal impact. The results of the proposed research will be made available in preprint form on open-access repositories such as Math arXiv. They will then be published in leading international mathematical journals. The PDRA and I will communicate our work to wide range of mathematical audiences at national and international conferences and seminars. We will seek to deliver lectures to broader public audiences.

A key feature of the grant proposal is the funding of a postdoctoral position (PDRA) to work on the problems outlined in the research proposal. This will give the PDRA the unique opportunity to learn the underlying theory and the methods of the proposed research project outlined in the Case for Support. He or she will undoubtedly grow and become an established researcher in the fields of Geometry Group Theory and Topology acquiring new knowledge and techniques by conducting research, writing scientific articles and developing communication skills by giving research talks and presentations.

There are rapidly growing applications of non-positively curved groups and spaces in Biology, Robotics, and Computer Science. On the other hand, (almost-)crystallographic groups, (almost-)flat manifolds and orbifolds are fundamental objects in Crystallography, Particle Physics, and String Theory. In the proposed research we will address some fundamental questions regarding these groups and spaces such as constructing computationally good models for classifying spaces of groups for proper actions and investigating the existence of Spin structures on almost-flat manifolds. From the first part, there are foreseeable applications to other more applied branches of Mathematics via the Isomorphism Conjectures. In the second, we study the existence of Spin structures on almost-flat manifolds. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. Therefore, there are potential benefits of the proposed research to physicists, biologists, and computer scientists.

As most of the envisioned impact of the grant proposal will be long-term in nature, we will focus on the initial societal impact. The results of the proposed research will be made available in preprint form on open-access repositories such as Math arXiv. They will then be published in leading international mathematical journals. The PDRA and I will communicate our work to wide range of mathematical audiences at national and international conferences and seminars. We will seek to deliver lectures to broader public audiences.

A key feature of the grant proposal is the funding of a postdoctoral position (PDRA) to work on the problems outlined in the research proposal. This will give the PDRA the unique opportunity to learn the underlying theory and the methods of the proposed research project outlined in the Case for Support. He or she will undoubtedly grow and become an established researcher in the fields of Geometry Group Theory and Topology acquiring new knowledge and techniques by conducting research, writing scientific articles and developing communication skills by giving research talks and presentations.

## People |
## ORCID iD |

Nansen Petrosyan (Principal Investigator) |

### Publications

Lutowski R
(2018)

*CLASSIFICATION OF SPIN STRUCTURES ON FOUR-DIMENSIONAL ALMOST-FLAT MANIFOLDS*in Mathematika
Lutowski R
(2019)

*Spin structures of flat manifolds of diagonal type*in Homology, Homotopy and Applications*Spin structures of flat manifolds of diagonal type*in Homology, Homotopy and Applications

Nansen Petrosyan
(2020)

*Cohomological and geometric invariants of simple complexes of groups*in Algebraic and Geometric Topology
Nucinkis B
(2018)

*Hierarchically cocompact classifying spaces for mapping class groups of surfaces CLASSIFYING SPACES FOR MAPPING CLASS GROUPS*in Bulletin of the London Mathematical Society
Osajda D
(2018)

*Classifying spaces for families of subgroups for systolic groups*in Groups, Geometry, and Dynamics
Petrosyan N
(2020)

*Bestvina complex for group actions with a strict fundamental domain*in Groups, Geometry, and Dynamics
Petrosyan Nansen
(2017)

*Bestvina complex for group actions with a strict fundamental domain*in arXiv e-prints
Prytula T
(2018)

*Solvable Subgroup Theorem for simplicial nonpositive curvature*in International Journal of Algebra and Computation
Prytula T
(2018)

*Hyperbolic isometries and boundaries of systolic complexes*in Journal of the London Mathematical SocietyDescription | The two main objectives outlined in the research proposal have been met. Together with collaborators Prof Andrzej Szczepanski and Dr Rafal Lutowski, we found all four-dimensional almost-flat manifolds that do not have a spin structure. This is a consequence of our explicit calculations where we list all possible spin structures for such manifolds. Regarding the second objective, together with the PDRA Tomasz Prytula, we showed that for any finitely generated Coxeter group the Bestvina complex is a classifying space for proper actions of minimal dimension. In fact, our results apply more generally to fundamental groups of strictly developable simple complex of finite groups. |

Exploitation Route | Building on our results on the Bestvina complex, Tomasz Prytula and I are working on simple criteria that will give new counterexamples to Brown's question as well as ways to compute Bredon cohomological dimensions of groups in this setting. This groups will be fundamental groups of strictly developable simple complex of groups. In relation to spin structures on almost-flat manifolds, it would be worthwhile to define a Dirac operator for some of these manifolds and investigate its spectrum. |

Sectors | Other |

Description | Spin structures on almost-flat manifolds |

Organisation | University of Gdansk |

Country | Poland |

Sector | Academic/University |

PI Contribution | I have studied classification of spin structures on almost-flat manifolds and found connections between the classifying map of the tangent bundle and the holonomy group of the manifold. |

Collaborator Contribution | Prof Andrzej Szczepanski and Dr Rafal Lutowski brought expertise on almost-flat and flat manifolds. Their knowledge of the flat case was vital for the project. |

Impact | So far, there are two publications that resulted from this collaboration "Classification of spin structures on four-dimensional almost-flat manifolds" and "Spin structures of flat manifolds of diagonal type". |

Start Year | 2017 |

Description | Classifying space for proper actions for groups admitting a strict fundamental domain |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Professional Practitioners |

Results and Impact | This was a research talk by Tomasz Prytula (PDRA) aimed at Geometric Group Theory research team members in IMPAN, Warsaw, Poland. Tomasz discussed our research results related to the EPSRC grant. |

Year(s) Of Engagement Activity | 2017 |

Description | Classifying spaces for families for non-positively curved groups |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Professional Practitioners |

Results and Impact | This was a research talk by Tomasz Prytula (PDRA) aimed at Pure Mathematics group in the Department of Mathematics at University of Southampton. The audience consisted of 10 permanent staff and 8 postgraduate students. Tomasz discussed his research interests and results related to the EPSRC grant. |

Year(s) Of Engagement Activity | 2017 |

Description | Dimensions of discrete groups and Brown's question |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Professional Practitioners |

Results and Impact | This was a research talk by Nansen Petrosyan (PI) given in the Algebra Seminar at University of Glasgow. Nansen Petrosyan discussed research results on Brown's question. This is related to one of the objectives of the EPSRC grant. |

Year(s) Of Engagement Activity | 2017 |

Description | Graphical small cancellation theory and simplicial non-positive curvature |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Professional Practitioners |

Results and Impact | This was a research talk by Tomasz Prytula (PDRA) aimed at Geometric Group Theory research team members in the Department of Mathematics at McGill University, Montreal, Canada. Tomasz discussed his research interests and results related to the EPSRC grant. |

Year(s) Of Engagement Activity | 2017 |

Description | Presentation about crystallographic patterns and tilings at the M4TH5 W33K3ND at the Winchester Science Centre |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Public/other audiences |

Results and Impact | Presentation about crystallographic patterns and tilings with a poster, games and puzzles at the Maths Weekend at the Winchester Science Centre, 12-13 May, 2018, designed for school children and the general public |

Year(s) Of Engagement Activity | 2018 |