 This lecture is part of an online course on lead groups and will be about the Bianchi classification. So the Bianchi classification is a classification of lead groups or algebras of dimension at most three. Bianchi did the three-dimensional case. So you recall the idea for classifying lead groups as as follows. First we classify the lead algebras. Then we find the centre of these lead algebras. And the lead groups is just the simply connected group divided by something in the centre. Some discrete subgroup of the centre. So once we found the lead algebras the groups are fairly easy to find. So the main problem is finding the lead algebras. And we're going to do, well obviously you can't do the disconnected ones so I should really have said this was connected lead groups. So let's just do the cases of dimension less than three. These are quite easy. So dimension naught is trivial. There's only one connected group and one dimension, one lead algebra. Dimension one, the lead algebra. Well there's only one possible lead algebra which is just a one-dimensional vector space with bracket zero. The simply connected group is the real numbers under addition. And a discrete subgroup of it is either trivial or isomorphic to z. So the only other possibilities we get are modulo z which is the circle group s1. In dimension two the lead algebra is spanned by two elements a and b. And the only non-trivial bracket is that the bracket of a and b must be equal to something. And we may as well choose a basis so the something is equal to b. I mean if it isn't we just choose a different basis. So we may as well take ab equal to b. All the other possibilities of course it might be zero in which case we can't take it as part of the basis. So there are exactly two possibilities. The lead algebra can either be r squared with the trivial bracket or it can be the lead algebra of this one which you remember as the lead algebra of the ax plus b group. You can think of this lead algebra as consisting of all matrices ab naught zero zero. And the corresponding groups, well here there are three essentially different groups because we can take r squared modulo zero which is just r squared or r squared modulo or discrete group isomorphic to z or r squared modulo z squared. And for this one the corresponding group is the matrices ab naught one with a greater than zero and this group is no center so there are exactly four lead groups of dimension two optoisomorphism connected ones. So now we'll do dimension three. I should mention that these are closely related to the Thurston geometrization conjecture which was proved by Perlman which says that every three-dimensional or every compact three-dimensional manifold can be chopped up into pieces and each of these pieces is one of eight geometries. And these eight geometries are quite closely related to several of the three-dimensional lead groups. In fact seven of the geometries can be represented as left and variant metrics on these groups and there's one geometry left over which doesn't quite fit in. So it's not an exact correspondence unfortunately. So let's see how to classify the three-dimensional lead algebras. So let's take a lead algebra l of dimension three over the reels. Then we can ask does it have, does l have a two-dimensional normal sub-algebra? Well what is a normal sub-algebra? So let's call this normal sub-algebra m. Well we say m is a normal sub-algebra in l. If m, l is contained in m whenever m is in m and l is in l. This is very closely related to the concept of a normal sub-group. So for the simply connected groups a sub-algebra would be normal if and only if the corresponding sub-group was normal in l. So this corresponds to the notion of a normal sub-group. And first we'll assume that l does have a two-dimensional normal sub-algebra. In this case the two-dimensional sub-algebra m must be one of the two-dimensional algebras we classified. So it's either equal to r squared or it's equal to the ab algebra. And this means that l has an extra element x. So l is generated by say x and m. And m goes to the bracket of x and m is a linear transformation of the space m. Now in the case when m is equal to the ab nought group it's not too difficult to check that this linear transformation must be of the form m goes to x, m prime, sorry m goes to m prime m for some m prime in m. And by subtracting m prime from x we can assume x equals 0 in which case the Lie algebra is just the product of this two-dimensional algebra m by one-dimensional algebra. So that's not terribly interesting or well it's mildly interesting but there's not much to do. So we can assume that m is just r squared with bracket 0 and we can take x to be any linear transformation. So let's look at what the possibilities for the linear transformation of m are. And we can write these out quickly. First of all we can have the eigenvalues might be the same and they might be the same and 0. And if they're the same and 0 there are two possibilities up to isomorphism. We can take the 0 matrix or we can take a sort of little nilpotent matrix. Well they might be none 0 so the eigenvalues might both be 1 and well the linear transformation of m well we can rescale x so you can multiply the linear transformation by none 0 constants you may as well assume the eigenvalues are 1 and this again leads to two possibilities. We could have the identity matrix or we could have a sort of identity matrix with a 1 up there as in Jordan normal form. Alternative for the eigenvalues could be different and in this case the matrix has to be diagonalizable and in this case we could have 1 eigenvalue could be 0 in this case we can assume the matrix looks like this by rescaling it or something or the eigenvalues could both be real and both none 0 and in this case by rescaling we can assume that looks like 1a. There's actually a sort of special case when a is minus 1 which is a little bit special or finally they could be complex in which case the matrix could looks like a plus b i or a minus b i if you diagonalize it of course a real matrix doesn't look like this what I mean is if you think of it as a complex matrix you could turn it into this form and again a equals 0 is a particularly special case of this so we can essentially write down all the Lie algebras with a two-dimensional normal subgroup just by going through this list and working out what the corresponding Lie algebra is and let me go through these so first of all we can have the matrix is just 0 this is called type 1 in the Bianchi classification the Bianchi classification is divided up into types for some reason so here the Lie algebra has got a two-dimensional Abelian Lie algebra with something else acting trivial on it so it's just the Abelian Lie algebra R3 and just as before we know what the corresponding groups are the groups are R3 modulo 0 z z squared or z cubed so we just get four different groups corresponding to that then we get the matrix 0 1 0 0 and this is called the Bianchi group of type 2 and the corresponding Lie algebra is the Heisenberg algebra which we will be discussing fair amount later and you can think of this as being all matrices that look like this and the corresponding group well the simply connected group consists of all matrices like this and it's got a center which consists of essentially the things where this is none 0 and those two are 0 so the center is just a one-dimensional Lie algebra and there's octoisomorphism there's only one way to find a non-trivial discrete subgroup of the center so we can take this group or we can take this group and mod it out by the matrices of the form like that where we take some element of the integer so there are two groups here these two correspond to certain Thurston geometries, this one corresponds to the Euclidean geometry and this one corresponds to something called a Nil geometry so a typical compact 3-manifold with Nil geometry would you take the Heisenberg group with all real coefficients and quotient it out by the subgroup where these three entries have to be integers so next we come on to the matrix 0 1 0 0 which is the type 3 in the Bianchi classification and this is just the group we mentioned earlier it's just the product of the reals times the A B 0 1 group or we could do I mean the Lie algebra is almost the same except you put a 0 there and here there are just two interesting groups because you could replace this R by S1 this corresponds to the in the Thurston geometry it corresponds to 3-manifold sort of modeled on the hyperbolic plane times the reals then there's if you look at the matrix 0 1 0 0 this is the type 4 Thurston sorry the type 4 Bianchi group and it doesn't really correspond to any Thurston geometry and frankly I can't really think of anything very interesting to say about this group if you look at the matrix 1 0 0 1 this is the type 5 Bianchi group and for Thurston's geometries it corresponds to the hyperbolic geometry H3 in other words you can put a left invariant metric on this group which turns out to make it into a hyperbolic space if we look at the matrices of the form a 1 0 0 with a not equal to 1 these give you the type 6 Bianchi groups we notice there's an infinite family of such groups because you can take a to be anything bigger than 1 for example and that gives you an infinite number of non-isomorphic groups the special case when a is minus 1 is particularly important this is sometimes called type 6 sub 0 it's a bit of an accident that how these things were divided up into types I mean this should really have been broken off as a separate type it's actually corresponds to the two dimensional Poincare group in other words the group of isometries of a two dimensional copy of Lorenzian space or at least it's connected component in Thurston's classification of geometries this corresponds to the so-called Sol geometry Sol stands for solvable that this is a sort of solvable group I guess or maybe whatever and then finally we get to the matrices of the form a plus b i a minus b i when you diagonalize them these are called type 7 there's a special case type 7 sub 0 again it was accidentally not it was accidentally put in with this case when it's really something special this is where you take a zero and it corresponds to this matrix here well this is obviously the diagonalization of a rotation so these sort of correspond to the rotations cos theta sin theta minus sin theta cosine theta and in fact this three dimensional group is just corresponds to isometries of a Euclidean plane and the isometries of Euclidean plane is not actually simply connected it's got an infinite fundamental group so you can take an infinite cover of that and get quite a lot of different groups by taking quotients by something in the center so again that there are infinite families of these groups well that more or less gives a quick summary of the three dimensional groups that have a normal subgroup of dimension 2 now we get onto the groups which have no two dimensional normal subgroup these groups actually turn out to be simple the algebras so simply algebra means there's no normal subalgebra other than nought and the whole the algebra so this is the analog of simple groups which are groups which have no normal subgroup other than the obvious subgroups we'll probably be discussing the classification of simple algebras in general later so what I'm going to do is I'm just going to quote things from the classification first of all there's exactly one three dimensional le algebra over the complex numbers that is simple and it's just the le algebra s l 2 c which is 2 by 2 matrices a b c d with a plus d equals zero and there are two three dimensional le algebras over the reals that are simple that become this one if you tensor them with the complex numbers and these are the le algebras of s l 2 of r which are just like that and there's also the le algebra of s u 2 of r which is the le algebra of the special unitary group so corresponding groups would be the le algebras s l 2 of r and the group s u of r which is isomorphic to the group s 3 that we discussed earlier and this one is simply connected and it has a center of order 2 so we get exactly 2 real le groups which are this group here or this group here quotient out by the element of order 2 so we can quotient out by 2z and this turns out to be the special orthogonal group in three dimensions of dimensional rotations as I mentioned earlier this group is not simply connected it has a center that's isomorphic to the integers so we can take its universal cover which is sometimes noted by s l 2 with a twiddle over it and this is center isomorphic to the integers so we can quotient out by any subgroup of the integers and get a infinite discrete family of groups with this le algebra so these correspond to Thurston these are the Bianchi groups of type 8 and 9 and they correspond to two Thurston geometries this is called the Thurston geometry of the universal cover of s l 2 r because no one could think of a better name for it and this is called spherical geometry this is particularly notorious because one obvious example of a manifold with a spherical geometry is the manifold s 3 and the Poincaré conjecture asks whether this is the only simply connected compact 3 manifold and this was finally answered by Perlman after about a century of attempts at it so that gives the more or less gives the classification of three-dimensional connected le groups I should finish off by saying there's one other Thurston geometry there's an extra Thurston geometry based on s 2 times r which very annoyingly doesn't correspond to any three-dimensional le groups so there's sort of overlap between Thurston geometries and Bianchi groups but they're not quite the same as each other I'll just say briefly what happens in dimension greater than or equal to 4 what happens is the simple le algebras and groups can be classified there are only a finite number in each dimension these were classified by killing over the complex numbers and by carton over the reels and I plan to discuss this classification in some later lectures the solvable ones just turn out to be a horrible mess so if we look at the classification of solvable ones in three dimensions what you see is that we get six or possibly seven or eight different types and they're beginning to look like a bit of a mess because there are rather a lot of two by two matrices and as the size of the the algebra goes up the complexity just goes completely wild and you can sort of push the classification to dimension four or five I'm not quite sure how far people have got but beyond about six it just gets hopelessly complicated to do a general le algebra over the reels I'm talking about finite dimension le algebras it turns out to have a normal subgroup that's solvable and if you're quotient out by this normal solvable subgroup you get a product of simple le algebras so the product of simple le algebras that sits on top is sort of classified by killing and carton and the solvable normal sub-algebra is as I said these are just horrifically complicated in high dimensions a typical example of this might be the set of all the le algebra of all matrices that look like this so what we do is we just take all five by five matrices and we set this block equal to zero and then the solvable normal solvable sub-algebra sort of consists of this block of six entries together with diagonal the diagonal matrices here with equal eigenvalues and the diagonal matrices here with equal eigenvalues and the product of simple le algebras we have a SL2 here of the trace zero matrices here and an SL3 of the trace zero matrices here so this is in some sense a moderately typical example of a le algebra in high dimensions it's got a solvable thing with a product of simple things sitting on top of it ok that would be enough about three dimensional le algebras for the moment