"We request that you and your fellow organizers devote an hour or so of the conference's time to a session on the state of the art, current directions, and emerging opportunities in the fields represented. The discussion should form the basis of a 'white paper' on the subject. The white paper should be made available to the conference participants as well as to the Division of Mathematical Sciences at NSF. This is one of the ways in which a well-organized conference can multiply its effectiveness. In addition, these white papers are an important source of information for the division when it makes its reports within the National Science Foundation and to the congress."I asked all the main speakers at Albany Group Theory Conferences, both this one and the previous ten, to form the `panel' and opened the floor for discussion. A number of other individuals in the auditorium contributed as well. What follows is a lightly edited transcript of the proceedings, followed by a summary with comments by the organizer. Due to the very unstructured nature of the `round-table', the discussion began rather slowly.
G.A. Swarup One thing that I find very interesting which was done recently is what Panos (Papazoglu) has done about splitting over virtually cyclic groups. I think there are new techniques there.
Gilbert Baumslag Chuck (Miller, in his talk) pointed out that we've moved from one area to something that seems very geometric. A lot of the talks at the conference were geometric---all but Olga's, which was also filled with geometry. The geometry seems to have taken over---what one might call `primitive combinatorial geometry', that is fooling around with diagrams and pretending that you have a proof, so to speak.
John Meier Do you view this as a positive thing?
Baumslag This is just a remark - not a judgement but an observation. It seems like primitive combinatorial geometry. People draw pictures in the plane and tell you that it is in n dimensions and you are supposed to make a deduction from the picture. I don't know whether it's valid or not. I think that Carl Ludwig Siegal once said that at least 90% of math papers contain errors and probably 50% contain errors that cannot be fixed. I wonder whether this geometry is like the geometry of the 19th century, the algebraic geometry of the Italians, where they did much that was incorrect that was later repaired with the influx of commutative algebra. Much of it was made valid by algebraic techniques. (I don't know if that is true or not.)
Swarup It was probably due to a lack of time - 50 minutes - to draw diagrams and sketch proofs.
Mark Feighn The techniques, topological and geometric, that are being used are well established.
Baumslag How rigorous are the proofs using Van Kampen diagrams? That's not the state of the art, but the unstate of the art or the state of the unart.
Swarup I never understood Van Kampen arguments.
Dani Wise I'd like to respond to what Gilbert said. There isn't actually a gap between the topologists and algebraists. Several years ago Bill Menasco gave a beautiful talk, and he explained that algebraists were novice topologists.
Marshall Cohen I have two comments: first a general one. I'm surprised at Gilbert's comment because it dawned on me `what field is it that we are asking the state of?'---I thought we were asking about the state of geometric group theory and I thought that that was what this conference was, actually. What field is it that we are talking about? I think that there is a continuum, obviously, of topological to geometric to purely combinatorial methods and I thought we mix them all up. I thought the talks using diagrams were quite rigorous. That's the general comment, that it is worth discussing what field we are talking about. On a technical detail about the question of Van Kampen diagrams: the short talk I gave this time was based on a fact started a year ago in the Cornell seminar (Dani Wise made everybody talk) so I had to find something to talk about and I decided that the thing that always confused me and that I didn't believe in (and still don't believe in some ways) is Van Kampen diagrams: there are several views of Van Kampen diagrams, but there is one extremely rigorous view which I ran into and which got involved in the particular stuff I talked about. This view where you take handlebodies and then take the dual graphs---that's really precise and I can understand those proofs. Those kind of diagrams, sometimes called pictures, I don't know how they relate to standard diagrams as in the Lyndon-Schupp book, those I think are really precise. But I think that is relatively minor and technical, compared to the larger questions that everybody will talk about.
Panos Papazoglu So maybe I should defend Van Kampen diagrams. Since I have been using them, I know that there are many mistakes in the literature, but as far as I am concerned, there is no problem---and if you have any problem with them, come and talk with me afterwards.
Baumslag I wasn't suggesting that there was a problem---just commenting.
Turner May I ask a leading question here? Is there one theorem that any one of you would like to prove in the next several years, that you think would be the most exciting result in the area---or a complex of issues to be resolved?
Baumslag That's a very difficult question. One of the things that strikes me is tangentially how little effect the extraordinary rise in computer power and computer storage has had on the community. You can see elements in the talks of the younger people who seem to have the ability to do more than one thing, and do exploit the computer, and I think that in ten or twelve years there's going to be quite an alteration in the kinds of talks that are given. So the state of the art has yet to be attained. It does look as though the computer has yet to have a profound effect on what it is that we do, even if we don't know what we are doing. You can see touches of it in some of the talks that were given. When it actually comes to working out an example which is in any way complicated without a computer, forget it---or so it seems. (Whose idea was it to have this silly round table?)
Jon McCammond I have a few things to say about some of these issues. As far as the field that we are in: geometric group theory relates to some neighboring fields that we don't usual pay much attention to, like combinatorics. A lot of the time, we focus on what's gone on in classical differential geometry and really try to imitate that and to pull their results down to our level, and we sort of view ourselves as making things discrete and doing really concrete, not analytic things. But if you talk to combinatorialists and you tell them a definition like CAT(0), they'll view you as an analyst. They don't think that you're doing combinatorics, they think you're still doing analysis because you talk about geodesics and you talk about an arbitrary path. The thing that they are really interested in doing, and there's a number of people working on this, is trying to mimic differential geometry but trying to do all of it combinatorially and so there are people who have done papers on combinatorial Ricci curvature, combinatorial vector fields, combinatorial differential manifolds---they are trying to pull all tools from differential geometry down into combinatorics and the way that I try to sell geometric group theory to them is that we are the halfway station, that we can pull things from arbitrary differential geometry down into piecewise constant curvature things and then they can pull them further down into combinatorics and I think that we need to view ourselves as part of a larger project that is going on in a number of fields.
Bob Gilman My question has to do with automatic groups, which were introduced a number of years ago, I guess as models of the fundamental groups of the known compact three dimensional manifolds, and so have a connection with low dimensional topology and I just wonder whether the theory of automatic groups has been a success - whether there IS a theory of automatic group?
Baumslag It's a wonderful theory with extraordinary ramifications, some of which have yet to be realized. The idea, as Thurston puts it, of actually having a group in which the only way you can multiply elements is by a computer is absolutely fascinating. The fact that nobody knows where to go with it is part of life---wait five or ten years and see. Maybe nowhere maybe somewhere, but it is in some sense a harbinger perhaps of things to come, because in many instances if you actually want to compute something it's very hard, even with homeomorphisms of some space that is sufficiently complicated---how do you combine them and how do you represent the answer? So actual working out of examples is very difficult---automatic groups is such that the actual multiplication is done by machine---you can't multiply two elements if the group is at all complicated. So to answer the question: I think it's a fantastic success.
Walter Neumann I think that my answer to Bob's question would be that if you look at automatic groups alone that they may not have been as successful as one might wish but they are a paradigm for a much larger theory and a way of looking at groups which I think is very successful. Putting extra algorithmic structure within the group, one can adjust the structure one looks at, but one should not stick only with automatic groups: the general ideas are very useful.
Ilya Kapovich I have to agree with Gilbert and Walter and think that in a certain sense our lives are going to get more difficult and more interesting in the years to come because the connection between group theory, topology and geometry is always going to be there but I think some other connections will become more prominent and it will not be enough to have algebraic and geometric intuition: you will also learn to think as a computer scientist or a statistician or someone who does probability theory. I think that in particular, those computational ideas will become more important and automatic groups is just one sign of things to come. I think that geometric group theory, like an ameba, is going to stretch its tentacles and touch other areas that we don't think about much yet. We will need to become even more encyclopedic that we are now. Be prepared.
Swarup I think there is one small concrete instance where it has influenced low dimensional topology. Earlier we looked at all subgroups as the same. Now we look at rational subgroups and try to see if they have LERF property. This seems to be that sort of influence at the technical level. This seems to be a definite influence on low dimensional topology.
Cohen I think that the ramifications for how we educate grad students are very serious. I agree that the computational issues are important (although I don't know anything about computing)---and that's part of the point I'm trying to make. I think the younger people, perhaps, are stronger and getting better educated in computer science, but one thing that I also think is happening is that a lot of the older people in the field have their education in the topology of manifolds and algebraic topology and you often don't see that in new graduates---they have grown up in a field created by older people who don't know what they have created: `my God, what have we done?' I think that the call on how we have to educate students now is very broad; I think that geometric background and deep geometric understanding is extremely important for giving a global picture of what one is doing and some taste in certain types of problems. But on the other hand there are all the other things, that area just mentioned, the range of stuff---so I think that the educational problems should really be thought about.
Meier Since educational issues have been brought up, I am going to be the representative of the liberal arts colleges and the undergraduate education. I actually think that we are at a point where I think we need to start getting a lot of geometric group theory ideas all the way down into the undergraduate curriculum. I don't see any reason why an undergraduate should take an abstract algebra course and know lots of things about Sylow theory and not have heard about the word problem. It seems to me very accessible---I've had lots undergraduate students work in geometric group theory---they can get a hold of some of these ideas, and they are going to know something current and they can see lots of places where fields start interacting, and that's exciting.
Boris Goldfarb I'd like to say something that is related to both of these comments. I was a student at Cornell studying topology of manifolds, but I never missed any group theory seminar (which was kind of combined ) which helped me with problems in topology---the Novikov conjectures---that depend on a geometric point of view on groups. Also, I should mention, in algebraic topology there is the work of Fred Cohen who calls his work `combinatorial group theory in algebraic topology'. What he does is he looks at homotopy and cohomology groups that come up and studies them using combinatorial methods and even Van Kampen diagrams, I think. So there is an interaction that is not just one way. My point is that when you educate, the geometric group theory is it's not just an illustration of whatever abstract group theory that you teach but it's useful techniques that students can use, maybe even undergraduate students.
Phil Hotchkiss I'd like to follow up on an earlier question. I'm at a small college, even smaller than John's, and I always like to tell my students what some of the big ideas are in mathematics even though they can't grasp all the subtleties of it. So I'd like to know what everybody thinks are the big questions in geometric group theory, what are the problems that are going to drive the field in the next five or ten years.
McCammond I'll reiterate what I said before, that we are part of a larger project of creating a completely discrete theory of differential geometry, and that I think that some of us in geometric group theory think of ourselves as inferior to differential geometry people but I think that we can play the opposite role of that kind of interaction with the combinatorialists.
And partly what I am thinking of here is from talking to combinatorialists---a number of times they create a partially ordered set and they look at what they call the order complex (the geometric realization) they do topology on it and they get a lot of results that look like results that should come from curvature. But they are never putting metrics on their complexes. And it could be that you can go and look at their field and say `we put these metrics on their complexes, apply CAT(0) ideas, apply all the curvature ideas, all the stuff that we have, and end up with very simple proofs of some of the theorems that they have done in the past'. I think that there are a lot of places where you can bring differential geometry down to geometric group theory and then port it all down to combinatorics. And that the geometric insight that you bring ends up having a lot of added value for them.
Swarup I don't find geometric group theory as exciting as it was in the late 80's. When Gromov started, there was a lot of excitement. It seems that at the moment there is a lull, and some figure like Gromov should come and show the new directions now---it seems a lot of scattered things are going on. Many of the major exciting things are over too quickly. Somebody again should set up a program : what are the problems? what are the projects? what are the directions to look at? I found that really, once Gromov lost interest in it, there doesn't seem to be any major figure showing the directions now. I don't think the field is as exciting as it used to be.
Neumann I think that the answer to that is that we should have the young people be down here (at the `round table') instead of the old.
Feighn One measure of the health of a field is its problems, another is the young people it attracts. By both measures, geometric group theory is healthy. I point to the problem lists maintained by Mladen Bestvina and the Magnus group, and to the many good results produced by the younger generation.
Baumslag I think that the field doesn't actually have any specific major problems and it's not like some other fields where people could say, `well let's solve Fermat's last problem', or something like that. It's a field which has been sort of broadly developed over the last twelve or thirteen years, and when it started, Gromov was such an extraordinary man, that he provided a lot of excitement and interest and there were, in fact, a whole body of people who actually stared to work in the field. The field didn't exist before Gromov---but yet it actually existed a long time ago, it started with Dehn. So in some sense the field is now old, but it was never new, because it just wasn't. I think a lot of mathematics is like that---if you just look it like this you say `oh God, we're doing all this brand new stuff'. But if you go back in the history, you suddenly find out that `no, it's not new at all!'. Small cancellation theory, of which hyperbolic groups are now in a sense, the ultimate generalization, are old. So, I think that's the way that everything develops: you try something for a while, you get tired of it, and then ten years later, somebody else tries it and suddenly it becomes interesting again. It ebbs and flows like life itself.
How's that? I thought that was pretty good.
John Meakin From the point of view of a relative outsider (speaker has worked primarily in semi-groups) it's always seemed to me that the field has been a rich source of algorithmic problems and there have been lots of algorithmic questions and wonderful topological and geometric techniques and it's also seemed that in recent years there have been a number of influences from computer science type directions, finite automata and various other things. I'm wondering to what extent this is a real new influence in the field and if so to what extent issues like computational complexity become serious matters such as in recent work of Sapir and so on.
Chuck Miller I'd like to comment that I think that algorithmic issues are rather fundamental to all of mathematics and not particular to combinatorial group theory or geometric group theory---that there's a pervasive interest of mathematicians in algorithmic questions. And think that that is likely to persist and increase since computers are able to do things that we once thought were practically infeasible. So I don't think that's going to go away at all---in fact, I think that it's going to become obligatory for us to train students to do things like program computers. We don't insist on that now but I think that we should. On a related issue, the influence of probabilistic ideas, particularly in a computational setting are going to increase, and I think that there are some fascinating new developments coming in that way. Those are the two directions that I think we need to see.
Olga Kharlampovich I think that the influence of computer science really changes the subject, not only technically but philosophically. For example, it definitely changes the notion of an algorithm, so that probabilistic algorithms, genetic algorithms start playing an important role. Questions like decidability of a problem in average become important. I really think that the philosophy of mathematics is changing because of computer science.
Kapovich I have to agree with Olga's point and I think if you look at the history, it's clear that computer science has been slowly making its way into our thinking because at first the question was whether a problem was decidable or nor decidable, like the conjugacy problem, and then we understood that you have to look at a somewhat more concrete situation, hyperbolic groups or something like that, where you have an isoperimetric function, or something of that sort, where it's not just decidable but there is a reasonable or fast way of doing it and by now even that is not enough, you really have to think about how to do it. The distinction was quite clear in Jon McCammond's talk that the distinction is already important between a theoretical algorithm and a fast algorithm even more so now that there are all these genetic algorithms which are not even algorithms, so what are they? These algorithms which sort of work most of the time on average---we still want to do it and I think that it is right that we do it, I think that the next thing we should do to understand what it is that we are doing there. This is one of the ways in which the computational complexity way of thinking as already affecting us.
Alexei Miasnikov About algorithmic problems in group theory---it's not just that we will have more computer science, it affects us, but computer science is changing now intensively, right now---it's different than it has been, and algorithmic problems are also quite different; word problems, complexity, time complexity, polynomial, quadratic. Now it's interesting that there are no papers on average complexity of problems in group theory. You have a group and in the worst case it's undecidable or very complicated, exponential, but on average it could be linear. So that, I think, is a very important direction, and it's related to cryptography and many other things like that. So maybe genetic algorithms work because on average problems are easy. So I think that that is a good direction.
Swarup On a more technical side, I think one of the best developments of the last few years has been the work of Olga and Miasnikov and Sela. And a lot of it is still not available. I hope that it will become available so that the younger people can see what are the future problems in that kind of area. That seems to really open some new directions. As far as I know, a lot of the material is still not available. It seems to be an area where there are lots of potential problems which younger people can take up.
Kharlampovich By the way, in general non-commutative algebraic geometry, there are many people trying to do this---like Kontsevich is doing something like algebraic geometry in free group rings, there are many approaches and I think it is a good direction.
Swarup I think the subject needs leadership, somebody again coming and telling what are the problems and what are the directions, not writing the problems but telling others these are the directions---I think that's what Gromov did. That seems to be lacking at the moment.
Papazoglu There is this new work that seems to be really deep on Tarski's problem and the machinery goes well beyond Rips' theory. Rips' theory was the big development in the last five years; a lot of good results came out of it, like the isomorphism problem for hyperbolic groups. And now we see that this theory goes one step further. I don't know what this means and I don't have an idea of what this could be. Maybe, for example, there is a way to use this technique to approach three-dimensional topology from a group theoretic point of view.
Swarup Perhaps one of the originators can tell us more about what are the future prospects.
Miasnikov Again about the influence of computer science. It turns out the Razborov-Makanin method is a big theme in computer science. They have separate conferences running on this subject it turns out; I didn't know but a lot of such conferences in computer science are related to the unifying problem in computer science, unification of terms. And recent developments mentioned by Olga, improvements of the Razborov-Makanin algorithm by Plondovsky and Diekert, they really tried to figure out what the complexity is, and they really simplified many pieces there. And, what I liked a lot, they used so-called `compression-data method', the people from computer science; they compressed data and then used it. But if you look at what they did, actually, they used old Lyndon's groups, Lyndon's completions, so compression means that you use parametric words in some special case. So it's really interactive with group theory. I think that Razborov-Makanin is the dynamical systems part of the JSJ decomposition, it seems like an effective side effect. I think it's very interesting.
Gabe Rosenberg I was wondering, along this line, if there might be too much focus being spent on how fast an algorithm might be as opposed to how much insight it gives us into the problem to begin with.
Meier I think my initial reaction would be that it would be difficult to separate the two. In some sense, figuring out and understanding that you've got an algorithm that runs this fast ought to mean that you have some insight into how your group is put together.
Papazoglu I would like to go back to Bob's question on automatic groups. There is one more related subject that seems to have disappeared, and that is isoperimetric inequalities. There was much talk about them and there is less interest now. And I think one reason is that one would expect some positive theorems and instead we got some counter-examples. Well actually, there was not much evidence to expect positive theorems, I think. But there is a general question, which has been interesting to me from the beginning and is still interesting, which is what distinguishes Cayley graphs (or spaces that admit a co-compact group action) from general metric spaces. I think that what happened in isoperimetric inequalities is one got the question and then wondered if these things are different in Cayley graphs from metric spaces, which is not a good idea. Nothing happened, there is no distinction. But maybe after all there may be something that distinguishes Cayley graphs from general metric spaces. There's Gromov's theorem on polynomial growth and there is actually another theorem of Varopoulos on isoperimetric inequalities that seems to be not very popular among group theorists, but I think that it is a theorem of this type which tells you that the Cayley graph is not just any metric space, it has some really different geometric properties. Unfortunately I have no conjectures to make, but I wonder if one can find natural properties that distinguish Cayley graphs from general metric spaces.
Wise I'd like to make a general comment on the `state of the art', which is in some sense a response to what Swarup has said. During many previous years, this conference has been overtaken by one theme or another, and there were many people who all seemed to be speaking about the same thing. This year, at this conference and at other conferences in the field that I have attended, there has not been one overwhelming theme. In fact, there has been a greater variety of viewpoints than we have ever seen before. These viewpoints and methods are engaging with each other and great progress is being made on many fronts. I do NOT believe that the field needs a single mathematical giant right now to tell us all what to do. There is a tremendous amount of talent, particularly researchers who are ten to fifteen years out of graduate school who are at the peak of their mathematical careers, and who are taking the field in many complementary directions. It was extremely useful to have a strong leader and visionary to get the field moving and to attract numerous talented younger mathematicians. But right now, a strong leader would have the wasteful disadvantage of setting too many people thinking about the same problems with the same viewpoint. We are currently tackling many different issues from many different vantage points. This is strong testimony to the health of geometric group theory.
Kapovich I agree with Dani's comment. Now is not the time for another prophet like Gromov to come and push us in one direction. We are in a very good state like a tree that is still growing rather bushily in various different directions---are we to bring an axe man and cut off all the branches except one? I'm not sure it's such a good idea right now, I think we have to wait, and grow in different directions for a while.
Swarup Time for conservation.
Baumslag Also time to stop.
There were several themes to which the conversation returned regularly, formulated below as questions. Each question is followed by a brief summary of the discussion. (The italicized comments are mine.)
1) Just what is our area?
The answer `geometric group theory' was suggested. The area of group theory as represented at the Albany Group Theory Conferences has certainly become increasingly dominated by geometric ideas, as has the general area of discrete group theory. Although there is some skepticism in some quarters as to the validity of some of the techniques (notably Van Kampen diagrams, which are more accurately topological than geometric), the general feeling is that they are not only sound, but extremely useful. Although in a sense the ideas are new, having been outlined at length by Gromov in the late 80's, they really originate in the work of Dehn (and Nielsen) in the early 1900's.
It would be short-sighted, however, to think of our subject as just `geometric group theory'. As the discussion and the contents of the talks clearly show, ideas of formal logic, computability and combinatorics as well as classical combinatorial group theory, combinatorial topology and homological algebra are all essential. One of the exciting features of this subject is that is has deep connections with many other areas. As discussed below, ideas of computer science are also becoming more important in our thinking. I will refer to our subject as just ` group theory'.
2) How do computers and computer science affect group theory?
This question has several answers. The first is that calculations that were at one time calculations `in principle' are now practical. This is illustrated by the MAGNUS website and the work of McCammond on recognition of low dimensional CAT(0) complexes. As pointed out by Miller, issues of algorithmic determination have always been central in mathematics.
On a deeper level, though, the incredible advance in computer power and computer storage is affecting the vary nature of what we consider important (and this will surely increase). The complexity of a solution is as important as the solution itself. Algorithms, like genetic algorithms, that work quickly most of the time but are slow in the worst case are interesting.
3) What is the role of automatic groups?
It is clear that the subject of automatic groups is less in vogue now than it was five years ago (as evidenced by the number of talks and papers on the subject). This may be due in part to the fact that instead of theorems that might have been expected, there have been counterexamples. Nevertheless, the viewpoint of automatic groups (in which the group operations can actually be done only on a computer) is fascinating and clearly important. Despite the recent lull in activity in this area, the notion of an automatic group is a central one, both in its own right and as a paradigm for a larger theory and way of thinking, in which extra algorithmic structure is included in the group structure.
On a related matter, it was pointed out that the related study of isoperimetric inequalities has almost vanished entirely. It was suggested, though, that the notion of isoperimetric inequalities relates to the central question of how one characterizes Cayley graphs among all graphs.
4) What role does discrete group theory have in the undergraduate curriculum?
It was pointed out that many of the ideas of discrete group theory are quite accessible to undergraduates (John Meier has had particular success in this regard) and are perhaps more natural topics in abstract group theory courses than classical topics like Sylow theory. One advantage of discussing them is that students can be introduced to current developments, and that's exciting.
5) Do we need a new leader, like Gromov, to lead the way to future developments in group theory?
Everyone recognizes the fundamental role played by Gromov in showing the mathematical community how powerful the geometric viewpoint can be in discrete group theory. The program suggested by Gromov in the late 1980's has seen intense development in the last decade, and there is a sense in some quarters that the excitement has waned somewhat and that a new `guru' would be welcome. A more general sentiment, though, is that the field is very robust, `like a tree that is still growing rather bushily in several directions', and that this is a very good thing.
6) How do recent developments effect our training of graduate students? The set of topics which young group theorists must master now is rather different (and larger) than what it was 10, 20 or 30 years ago when most of the established people in the field were graduate students. Many of us (especially those over 45) have backgrounds in the topology of manifolds, differential topology and homotopy theory, for example, subjects that are much less commonly studied now. ( Perhaps this reflects the success of the leaders in those fields.) Many topics were important then as well as now - homological algebra, the logic of decision problems, basic combinatorial methods in group theory, the topology CW complexes, etc. But the areas of theoretical computer science and issues like complexity hardly existed 20 or 30 years ago, and the basics of geometric group theory (quasi-isometries and the sort) were essentially unknown. Furthermore, statistical and probabilistic ideas, although not new, have only recently been used in group theory. All these observations reflect on how we educate our students.
7) What is the role of statistical and probabilistic thinking. in group theory?
This is a direction that has seen very little development so far but can be expected to be very important in the future, particular with the explosion in computer technology. The genetic algorithms developed at CUNY are an indication of what might be expected.
8) How does group theory relate to other fields, particularly differential geometry and combinatorics?
A major motivation of geometric group theory has been an attempt to bring ideas of classical differential geometry into the realm of group theory (as witnessed, for example, by the study of hyperbolic groups and groups acting on CAT(0) spaces). But it was observed that there are also deep connections with developments in combinatorics. Perhaps our insights can inform the combinatorialists in much the way that the ideas of differential geometry have informed us. In any case, we should view ourselves as part of a larger project that is going on in a number of fields.