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# A Pointless Exercise

Sooner or later every professional mathematician has the experience of attending a party and getting in an awkward conversation with a non-mathematician about their research. Eventually, they always ask, What is the point? But no one ever asks *where* is the point. Sometimes this question is harder to answer than you might think.

If you are a mathematician and not a social psychologist, *group dynamics* refers to the study of group actions on spaces: we have a group a topological space , and an action of , on i.e. a homomorphism , where denotes the group of self-homeomorphisms of preserving some structure or other. We gain a lot of insight into the action by considering the orbits—i.e. how and its elements act on particular points: fixed points, periodic points, recurrent points, and so on. But occasionally it happens that the points *per se* are …well, they’re beside the point. We give three examples.

## 1. Mapping Class Groups

Sometimes we don’t really have a group action at all, but only a homomorphism where , is a quotient group of thus, we could think of elements of ; as *equivalence classes* of automorphisms of or (equivalently) that , only acts on *up to equivalence.* One of the most familiar examples is that of *mapping class groups*. If is a manifold, and we give the compact-open topology, path components in correspond to homeomorphisms that differ by isotopy (i.e. that can be connected to each other by a continuous one-parameter family of homeomorphisms). The path component of the identity is a normal subgroup, and the quotient is the group usually denoted , or sometimes .

The best studied case is when is a surface If . is oriented, we typically restrict our attention to the group of orientation-preserving homeomorphisms of (denoted and, by an abuse of notation, we let ) denote the group of orientation-preserving mapping classes. Given a subgroup of it is natural to wonder whether it comes from an “honest” action of , on i.e. whether there is a section from ; to The hardest instance of this question is the universal one: is there a section . In other words, is there some function ? taking mapping classes to representative homeomorphisms and satisfying for all ?

If is the 2-sphere, then is trivial, so this is easy. If is the torus, is and the linear representatives of each mapping class group constitute a section (after one fixes a basepoint and a flat structure on the torus). But in every other case the answer turns out to be ,*no*: Vladimir Markovic and Dragomir Šarić showed

This fact is not as disappointing as it seems: it is possible to turn into an honest group of transformations in a few different ways. If we pick a (any) Riemannian metric on and denote the bundle of unit tangent vectors to by then (rather surprisingly?!) there is ,*always* a section coming from the action on the *circle at infinity* of the universal cover of And there is a natural action of . on spaces that themselves parameterize homotopy classes of structures on such as the Teichmüller space of , or the complex of curves. ,

## 2. The Cremona Group

Sometimes we have a group action, but the transformations associated to group elements are not defined everywhere, and it might easily happen that the common domain of definition of the entire group is empty.

A *birational transformation* between two algebraic varieties is an isomorphism between Zariski open subsets of the domain and range. An example is the map between the open subset of the circle where , and the (affine) line. A birational transformation which is defined on the entire domain is a *birational morphism*. Birational transformations between smooth curves extend to honest birational isomorphisms between their projective completions. But in higher dimensions, the situation is more complicated. Birational transformations may be composed (after further restricting the domain), and the collection of all birational transformations from a fixed variety to itself is a group.

Let be a field, and let be homogeneous coordinates on The group . acts on by automorphisms in an obvious way. The involution

is an isomorphism on the open subset where no coordinate is zero. Near the point the map behaves rather badly: the line , (for is collapsed to the point ) whereas the line , (for is collapsed to ) The point . wants to be spread out over the entire line joining these two points, with the result that there is no way to continuously extend the domain of the function to all of The same situation occurs at . and at .

The group of all birational transformations of is known as the *Cremona group* and is denoted When . the Cremona group is generated by and but this is not true for general , (for example, for ).

Every element of may be represented by an honest birational morphism if we are prepared to first blow up the domain a finite number of times. The transformation can be promoted to an honest involution on blown up at three points in general position (Figure 1 shows the action of acting on blown up at three points), but for most elements of there is no finite blowup of on which acts as an honest automorphism.

The two-dimensional cohomology of is generated by the class of , topologically a 2-sphere with self-intersection number 1. If we blow up at a point , the preimage of , is an *exceptional divisor* an embedded 2-sphere with self-intersection number —topologically Yuri Manin .*bubble space* whose points are the equivalence classes generated by the relation if there is a birational morphism between two (iterated) blowups sending to and which is an isomorphism in a neighborhood of This .*space* is a little bit like the Bruhat-Tits tree of which one can think of as a kind of infinite fractal cactus, obtained from , by repeatedly attaching new copies of to every point.

Cohomology pulls back under blowup, and the direct limit of under all possible sequences of finite blowups is a free abelian group with one copy of for the generator of and one copy of , for every *point* in The intersection form on . has signature and the Cremona group acts by isometries of an infinite dimensional hyperbolic space (which lives inside a suitable , completion of Serge Cantat and Stéphane Lamy ).

## 3. Symmetries of Pseudoline Arrangements

A *pseudoline arrangement* is a collection of finitely many embedded oriented circles in so that each pair intersects transversely in exactly one point; it is *simple* if there are no triple points. We consider such arrangments up to the equivalence relation of isotopy. An arrangement of *straight* lines in general position is a pseudoline arrangement, but not every pseudoline arrangement is isotopic to a line arrangement (those that are are said to be *stretchable*). One example is illustrated in Figure 2, which would violate Pappus’s theorem if it were stretchable. A simple arrangement with nine lines is due to Ringel.

Peter Shor *degenerate* arrangement of lines (i.e. one with more coincidences) with the same symmetry. In fact, any finite group of symmetries of a pseudoline arrangement is isomorphic to a finite subgroup of .

It is slightly subtle even to define the symmetry group of an infinite arrangment; one way is as follows. A *marked* pseudoline arrangement of circles is a choice of bijection of the components with Let . denote the set of marked pseudoline arrangements; note that this is a finite set for any .

An * -cochain* on a group with *values in * is a function from distinct ordered of group elements -tuples to which is invariant under the (left) diagonal action of on Such a -tuples. is an * -cocycle* if there is some -cochain that *restricts to* (in the obvious sense) under all order-preserving inclusions from to Note that if . then the cocycle property for a , -cochain actually implies the existence of a (unique) -cochain for all that restricts to (in the obvious sense); i.e. a 4-cochain which is *compatible on 5-tuples* is *compatible on -tuples* for all Thus if . is a 4-cocycle on then for any finite subset , of there is an arrangement of pseudolines which is compatible with restriction and the natural (left) action of on itself. So, for example, if is a finite group, a 4-cocycle on gives rise to a pseudoline arrangement of pseudolines for which there is a natural permutation action of extending to all of .

If acts on by homeomorphisms, freely permuting the elements of some (possibly infinite) pseudoline arrangement, we obtain a 4-cocycle. Thus the 4-cocycle in a sense encodes *the* symmetries of some arrangement.

One cannot in general expect a *projective* action of i.e. a faithful representation , For instance, every group of (orientation-preserving) homeomorphisms of the circle acts by homeomorphisms of . with a common fixed point, permuting the set of straight lines through that point. At the other extreme, any homeomorphism of permuting *all* the straight lines is actually in .

## Author’s Note

The author would like to thank David Eppstein, Benson Farb, Seraphina Lee, Rich Schwartz, and the anonymous referees for their help.

## References

- [1]
- Serge Cantat and Stéphane Lamy,
*Normal subgroups in the Cremona group*, Acta Math.**210**(2013), no. 1, 31–94, DOI 10.1007/s11511-013-0090-1. With an appendix by Yves de Cornulier. MR3037611,## Show rawAMSref

`\bib{Cantat_Lamy}{article}{ author={Cantat, Serge}, author={Lamy, St\'{e}phane}, title={Normal subgroups in the Cremona group}, note={With an appendix by Yves de Cornulier}, journal={Acta Math.}, volume={210}, date={2013}, number={1}, pages={31--94}, issn={0001-5962}, review={\MR {3037611}}, doi={10.1007/s11511-013-0090-1}, }`

- [2]
- Yu. I. Manin,
*Cubic forms*:*Algebra, geometry, arithmetic*, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Translated from the Russian by M. Hazewinkel. MR833513,## Show rawAMSref

`\bib{Manin}{book}{ author={Manin, Yu. I.}, title={Cubic forms}, series={North-Holland Mathematical Library}, volume={4}, edition={2}, subtitle={Algebra, geometry, arithmetic}, note={Translated from the Russian by M. Hazewinkel}, publisher={North-Holland Publishing Co., Amsterdam}, date={1986}, pages={x+326}, isbn={0-444-87823-8}, review={\MR {833513}}, }`

- [3]
- Vladimir Markovic and Dragomir Šarić,
*The Mapping Class Group cannot be realized by homeomorphisms*, https://arxiv.org/abs/0807.0182, 2008. - [4]
- Peter W. Shor,
*Stretchability of pseudolines is NP-hard*, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 531–554, DOI 10.1090/dimacs/004/41. MR1116375,## Show rawAMSref

`\bib{Shor}{article}{ author={Shor, Peter W.}, title={Stretchability of pseudolines is NP-hard}, conference={ title={Applied geometry and discrete mathematics}, }, book={ series={DIMACS Ser. Discrete Math. Theoret. Comput. Sci.}, volume={4}, publisher={Amer. Math. Soc., Providence, RI}, }, date={1991}, pages={531--554}, review={\MR {1116375}}, doi={10.1090/dimacs/004/41}, }`

## Credits

Figure 1, Figure 2, and photo of Danny Calegari are courtesy of Danny Calegari.