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# Fibring of manifolds and groups

## Periodic Reporting for period 2 - FIBRING (Fibring of manifolds and groups)

Reporting period: 2022-01-01 to 2023-06-30

The central theme of the project is determining which groups and manifolds "fibre". A manifold is a space that locally looks like a familiar Euclidean space. So, for example, the circle is a 1-dimensional manifold, since it locally looks like a piece of a line; the surface of the Earth is a 2-dimensional manifold, since it looks locally like a piece of a plane; finally, the manifold that surrounds us seems to us to be 3-dimensional. Crucially for the project, the 3-manifold around us evolves in time, and hence the 4-dimensional space time can be viewed as 3-dimensions that dynamically change. This is precisely the idea behind fibring: a manifold fibres if it can be similarly thought of as a manifold of one dimension less, evolving in time.

Groups are algebraic objects that measure symmetry, similarly to the way numbers measure quantity. It turns out that a group can also fibre, which again allows us to view it as a simpler object evolving in time.

The overall objectives are to understand all the ways a given manifold or a group can fibre. Understanding this question will bring us closer to understanding all manifolds, including our own 4-dimensional space time. On the group level, it is important to have tools to comprehend a group at the ready, since all aspects of science that exhibit or utilise symmetries use group theory in a fundamental way.
The focus of the first half of the project was on studying fibring from the "finite" perspective, in multiple ways. One can try to understand a manifold by looking at manifold that are very much like the original manifold, but are bigger and exhibit additional symmetries, roughly in the way that a kaleidoscope picture arises from a single segment. We call this sort of a situation a "finite covering". One can looks at all possible finite coverings, and study a manifold this way, or one can try to find a finite cover with better properties than the manifold we started with. For groups the situation is similar, with finite covers being replaced by the analogous notion of a finite-index subgroup.

Studying properties that are determined by knowing only the extra bit of symmetry, but for all possible finite covers, is known as looking for "profinite invariants". We have shown that in a large number of situations, being fibred is a profinite invariant. We have also shown that a variety of groups and manifolds have finite covers that fibre, and in the groups case we obtained a characterisation of many different flavours of fibring using numbers associated to a group, called Betti numbers. We have also looked at how these Betti numbers behave when one decomposes a manifold in a way similar to a fibring, but in which the time parameter is allowed to be more complicated than a simple line.
All of the results described above go well beyond the state of the art. The work conducted so far had a transformative effect on our understanding of fibring. The concept is now much more widely used, and the underlying phenomena are studied more vigorously than ever before.

In the remaining 2.5 years of the project, we intend to focus on the dynamical aspect of fibring: the way the smaller space actually changes. One line of enquiry is studying the properties of various invariants associated to the dynamical system arising this way. We intend to focus on a analytic invariant known as a zeta function. The other aspect we want to understand is: which properties of the dynamical system stay the same for all the various ways our initial object fibred?