Thursday, July 21, 2005

Math Post the Eighth: Personal Interlude with Some More Definitions

So I am now officially a doctoral candidate! Heck yeah! Also I'm now (or will soon) be attending a lengthy algebraic geometry conference in Seattle, so I'm not even in Michigan anymore. Perhaps most importantly from a blogging perspective, I'm not madly studying every second of my day anymore (merely some seconds), so I should be able to get back to blogging on a more regular basis. So nyah, say I!

I've also redone my links section, doing away with links to friends' livejournals because I couldn't think of a good way to separate them from links to friends' more serious blogs or therefore from other peoples' more serious blogs. Any complaint and I'll rethink it, but it seemed like the right way to go at the moment.

However, I've decided to defer my promised discussion of isomorphism theorems and universal properties so I can, in celebration of my prelims, explain that portion of my research that I'm currently prepared to discuss (that is, that part that shouldn't be much over anybody's head for reasons other than density). So, in broad strokes, here we go:

One first kind of assumption that mathematicians like to make is a finiteness assumption. It's usually easier to deal with things that are "small" than things that are "big," and indeed strange things can happen in the world of large objects (they can be the same as several copies of themselves stuck together, for one). One finiteness assumption we've already used is precisely that: that the number of elements in a group is finite. A lesser one harks back to my discussion of generation of a group, in particular that our group has a finite set of generators, or is finitely generated. Many (though not quite all) of the examples I've mentioned in that first group post are finitely generated (an exercise for the reader to figure out which!). Making an assumption like this is more about choosing a field of study than picking out a special case.

A second definition I'll need is the condition of a subgroup having finite index. The index of a subgroup H in a group G is the number of cosets that you can form from H with elements of G. In the integers Z, the subgroup 3Z has three cosets, the subgroup 5Z five, and so on. For finite groups, Lagrange's theorem tells us that the index of a subgroup is simply the ratio of the orders (i.e. the number of copies of the smaller group it takes to cover the large one is the number of elements in one divided by the number of elements in the other). If we consider the group of ordered pairs of integers, the subgroup generated by the pair (1,0) is of infinite index, while the subgroup generated by the elements (2,0) and (0,2) is of index 4 (consider quotienting it out and note that you get a (the) two-element group in each component).

The first big assumption I'd make is that of being just infinite. A group G is just infinite if it is infinite but has only finite quotients. I'll cut to the examples in a second, but I need to add one more assumption to get to what I'm actually studying: I don't just want G just infinite, I also want every finite-index subgroup to be just infinite as well -- a condition called being hereditarily just infinite. Our first example is the integers. Since every normal subgroup (in fact, since the integers are abelian, every subgroup) has finite index, there can be no infinite quotients. A slightly more complicated example is the group of symmetries of an infinite line with equally-spaced dots on it (like the integer points in the real numbers), written as D_\infty, where the notation means an infinity symbol. These come essentially in two kinds -- shifts some number of spaces in one direction or the other or reflection around a particular point (or combinations of the two). The group actually looks like (something I'll get to later) the semi-direct product of the integers (the line movements) with the two-element group (a reflection about some point -- you can "make" reflecting about any point by translating, reflecting, then translating back). Seeing that this group is hereditarily just infinite is most easily done by thinking hard about group actions, which I will also get to later.

Before finishing up, I need one more assumption: that the group is not something I'll call simple. A simple group has no nurmal subgroups and therefore no interesting quotients at all. The finite groups of prime order are all simple; showing any other group to be simple (that is, actually doing the proofs) is well beyond the scope of what I'll be able to consider here for some time. Computing all the finite simple groups was a major research project, popularly considered finished around 1980, and filling many thousands of pages (so many in fact that people still aren't completely sure it's all correct -- there's a major effort to write it in some comprehensible way that's still going on). Infinite simple groups satisfy the condition of being hereditarily just infinite trivially (for the mathematicians in the audience, this is because any finite index subgroup H in a group G has a finite-index subgroup H' normal in G inside it -- think about it), but aren't very interesting examples for this same reason.

(For genuine group theorists, the previous paragraph is an ersatz for simply assuming G to be residually finite, which is equivalent for just-infinite groups but harder to explain)

The last class of examples is something harder to get your hands on -- for the big-word-friendly audience, I'm talking about irreducible lattices in higher rank semisimple Lie groups -- and harder still to show in general (again for the big-words crowd, this is Margulis' stuff). For the near future, you can simply think of them as big groups of integer matrices (say 3x3 or larger). My project is to show that these are the only interesting examples. It's likely to be uphill work, but it should at least lead to some intriguing stops along the way.


Anonymous Bill said...

Can you elaborate on the last paragraph a bit for those of us that are aquainted with the big words, but maybe not as friendly as some?

"Semisimple Lie Group" - got it. What is an irreducible lattice in this context. (I can probably figure this out but I am busy and lazy).

And, what exactly is the conjecture you are trying to prove. It looks like some classification theorem on hereditarily just-infinite groups, but I cannot figure out the details.

Congrats on your quals.

10:16 AM  
Blogger Dennis said...

Sure. A lattice in a Lie group is a discrete subgroup with finite covolume. A lattice in a product Lie group is reducible if its projections onto the factors are non-discrete. And yes, the conjecture is that every finitely generated hereditarily just infinite group that is not Z, D_\infty, or simple is isomorphic to an irreducible lattice in a higher-rank semisimple Lie group.

And thanks.

5:56 PM  

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