Interesting Mathematics

Existence of collisional trajectories of Mercury, Mars and Venus with the Earth

2010-10-07T07:01:00.001-04:00

Vadim Kaloshin was telling me about this article by Laskar and Gastineau. In 2500 numerical simulations of solar system evolution, one trajectory had other planets hitting the Earth - in a few billion years.

A Hollywood movie is confidently expected.

The Peter Principle Revisited

2010-10-02T09:03:00.000-04:00

Not really "coarse mathematics" in the sense I initially intended, but feels rather relevant to the work of a department head. This paper uses a simulation to demonstrate that, under certain hypotheses, a "hierarchical" organization will function more efficiently by promoting people at random than by always promoting those who are most competent in their current position.

"Mass endomorphism, surgery and perturbations"

2010-09-29T16:56:00.000-04:00

That is the title of an interesting paper just posted on the arXiv. I had never heard of the "mass endomorphism" so this was new to me... Take a compact Riemannian spin manifold and suppose that the metric is flat in the neighborhood of a point p. If there are no harmonic spinors (so that the Dirac operator is invertible) then the Dirac Green's function, i.e. the inverse of the Dirac operator, has an asymptotic expansion near p in which the zero term is an endomorphism of the spinor bundle called the mass operator. It is known that if the mass operator is non-zero then a solution exists to the classical Yamabe problem. In this paper it is shown that the mass operator is "generically" non zero - using a lot of the machinery from positive-scalar-curvature land: psc surgery, results of Stolz, etc...

Coarse Math at MSRI!

2010-09-16T10:08:00.001-04:00

Forwarded from Vincent Lafforgue

From August 15, 2011 to December 16, 2011, MSRI (Berkeley) hosts a program on Quantitative Geometry. It is organized by Keith Ball, Emmanuel Breuillard, Jeff Cheeger, Marianna Csornyei, Mikhail Gromov, Bruce Kleiner, Vincent Lafforgue, Manor Mendel, Assaf Naor (main organizer), Yuval Peres, and Terence Tao. This is a big program with many available positions.

Examples of areas that will be covered by the program are: geometric group theory, the theory of Lipschitz functions (e.g., Lipschitz extension problems and structural aspects such as quantitative differentiation), large scale and coarse geometry, embeddings of metric spaces and their applications to algorithm design, geometric aspects of harmonic analysis and probability, quantitative aspects of linear and non-linear Banach space theory, quantitative aspects of geometric measure theory and isoperimetry, and metric invariants arising from embedding theory and Riemannian geometry. Go to http://tinyurl.com/28x94y6 for more details.

The deadline for applications is
-October 1, 2010 for Research Professors
-December 1, 2010 for Research Members and Postdoctoral Fellows.
Look at
http://www.msri.org/propapps/applications/application_material
for more details.

Permanence properties in coarse geometry

2010-09-16T09:57:00.000-04:00

I hate to think how long it has been since I last posted here. My apologies - it has been a difficult summer for various non-mathematical reasons. Anyhow, trying to get back on track let me mention a survey article that Erik Guentner sent me called "Permanence properties in coarse geometry". What Erik means by "permanence properties" is statements like "the property of having finite asymptotic dimension is closed under group extensions". Many statements of this kind, for a variety of coarse properties (asymptotic dimension, embeddability in Hilbert space, property A/exactness, etc) have by now been proved and this is a very nice survey bringing together general techniques for obtaining such results with specific applications.

JohnR

Metric spaces with dilations, and metric trees

2010-07-15T07:30:00.000-04:00

Here is a gentle introduction to the theory of metric spaces with dilations ("rescaling maps", so that one can define an appropriate notion of tangent space.) This appears on the arXiv today.

http://arxiv.org/abs/1007.2362

Also a couple of elegant-looking papers on metric trees and their embeddings into Banach spaces

http://arxiv.org/abs/1007.2207

http://arxiv.org/abs/1007.2208

Lots of interesting stuff on the arXiv today. I probably won't be posting much for a few weeks as I have some personal business to take care of.

Smith theory made coarse

2010-07-06T15:32:00.000-04:00

"Smith theory" is the name given to the investigation (starting with the work of P.A. Smith in the late 1930s) of the homological properties of the fixed-sets for finite groups (especially $p$-groups) acting on spheres. See Dwyer, William G., and Clarence W. Wilkerson. “Smith Theory Revisited.” The Annals of Mathematics 127, no. 1. Second Series (January 1988): 191-198.

Ian Hambleton and his student Lucian Savin have just posted an article about a coarse-geometric counterpart of Smith Theory, Hambleton, Ian, and Lucian Savin. “Coarse Geometry and P. A. Smith Theory.” 1007.0495 (July 3, 2010). http://arxiv.org/abs/1007.0495.

The notion of fixed set is replaced by a "coarse fixed set", which is the coarse structure (if it stabilizes) of the sequence of "approximate fixed sets" $\{x: d(x,f(x))\le n\}$ for $n=1,2,\ldots$.

This notion is well behaved for finite group actions.

"Holomorphic Functional Calculus"

2010-07-02T18:25:00.001-04:00

Writing up the Connes-Renault notes, which I mentioned in a previous post, leads to a number of interesting digressions. For instance, the notion of "holomorphic closure" is discussed at some length in these early notes. But what exactly is the relationship between "holomorphic closure", "inverse closure", "complete holomorphic closure" (= holomorphic closure when tensored with any matrix algebra), and so on? I was aware that there had been some progress in this area but had not really sorted it out in my mind. Here's a summary (all these results are pretty old, so perhaps everyone knows this but me...)

Theorem 1. Let G be a group which has a complete metric topology for which multiplication is jointly continuous. Then inversion is continuous (i.e., G is a topological group).

There is a short, slick proof in Pfister, "Continuity of the inverse", Proc AMS 95(1985) 312-314, though the result is older.

For the rest of this note I follow Schweitzer, Larry B. “A Short Proof that $M_{n}(A)$ is local if $A$ is local and Fr\'echet.” International Journal of Mathematics 3 (1992): 581-589. Suppose that B is a C* or Banach algebra and that A is a dense subalgebra which is a Fr\'echet algebra under some topology stronger than the topology of B.

Theorem 2. A is holomorphically closed in B iff it is inverse closed.

For the proof, one notes that inverse closure implies that the invertibles are open in A, so their topology (in A) can be given by a complete metric. By Theorem 1, inversion is continuous. This means that the Cauchy integral formula for the holomorphic calculus converges in A.

Theorem 3. A is inverse closed in B iff every irreducible A module is a submodule of some B module.

This involves some topological-algebraic manipulation with maximal left ideals. An irreducible A module is of the form $A/m$, where $m$ is some maximal left ideal. Let M be the closure of m in B. Using density and inverse closure one sees that $m=M\cap A$. But then
$A/m = A/A\cap M = (A+M)/M \subseteq B/M $ is contained in the B-module $B/M$.

Corollary. $A$ is inverse (or holomorphically) closed in $B$ iff every matrix algebra $M_n(A)$ is inverse closed in $M_n(B)$.

This follows since an irreducible $M_n(A)$-module is just the direct sum of $n$ copies of some irreducible $A$-module.

Connes Embeddings and von Neumann Regular Closures of Group Algebras

2010-06-29T16:32:00.002-04:00

This is an interesting paper of Gabor Elek's which touches on some things I've posted about recently - especially (i) the Atiyah conjecture and (ii) the idea (which shows up in the work of Ara et al) that one can use some kind of "asymptotic rank" instead of "asymptotic trace" in some contexts where you want to build "continuous dimension" type invariants.

Exchange rings and translation algebras

2010-06-25T11:16:00.000-04:00

Recall that a ring $R$ is von Neumann regular if given any $x\in R$ there is $y\in R$ such that $xyx=x$. (Examples: fields; matrix algebras; various rings of unbounded operators, where $y$ is "the inverse of $x$ away from the kernel".) A ring $R$ is called an exchange ring if, for every $x\in R$, there is an idempotent $e\in R$ such that $e\in xR$ and $(1-e)\in (1-x)R$. Von Neumann regular rings are examples of exchange rings. There is a developed algebraic theory of these rings, with links to real rank 0 C*-algebras among other things.

I just became aware of the paper Ara, P., K. C O'Meara, and F. Perera. Gromov translation algebras over discrete trees are exchange rings. Transactions of the American Mathematical Society 356, no. 5 (2004): 2067–2079 (electronic). In this paper it is shown that the algebraic translation algebra associated to a tree, with coefficients in a von Neumann regular ring, must be an exchange ring. The tree hypothesis is used in various places, but the authors don't know, apparently, whether there are examples of metric spaces $X$ for which the translation algebra is not an exchange ring. (The plane might be a good example to start with.)

C*-algebras, foliations and K-theory

2010-06-23T17:53:00.000-04:00

In 1980, Alain Connes gave a course entitled "C*-algebras, foliations and K-theory". Jean Renault was a student in the course at that time and took notes, and photocopies of his meticulously handwritten manuscript have been passed around generations of students. I must have acquired mine some time around 1988.

The notes describe projective modules, Morita equivalence, K-theory, non-unital algebras and multipliers, quasi-isomorphisms, smooth subalgebras and "holomorphic closure", Bott periodicity, crossed products, the Thom isomorphism for crossed products, and the beginnings of noncommutative geometry. Its fascinating to see how early some of these ideas were germinating, and what they looked like at taht early stage.

In our seminar last year we assigned graduate students to read and lecture on various parts of the manuscript. This led to a rough English translation, which I'm now in the process of tidying up. I hope to post a more polished version to the arXiv before too long. I'm grateful to Alain and Jean for encouraging this project.

You can find scans (not too legible) of the lecture notes here.

Bounded groups

2010-06-16T16:43:00.001-04:00

Today's post arises from a paper by my Penn State colleague Dima Burago and his collaborators, Conjugation-invariant norms on groups of geometric origin. (October 7, 2007). http://arxiv.org/abs/0710.1412.

The basic definition is a very simple one: say that a group $G$ is bounded if it has finite diameter with respect to any bi-invariant metric.
(Of course it is enough to consider the distance from a group element $g$ to the identity element, and this function $|g|$ then becomes a \emph{conjugation invariant norm} on the group, hence the title of the paper. A group is then bounded if every conjugation-invariant norm is bounded from above. One can also ask whether every conjugation-invariant norm is bounded from below (away from the identity element). When a norm is bounded from above and below and can be said to be trivial: it is equivalent to the norm which equals 1 on all non-identity elements. If every conjugation-invariant norm is trivial the group is called meager.

The basic phenomenon studied in the paper is the surprising fact that conjugation-invariant norms on large groups of diffeomorphisms tend to be trivial. In fact the main result is that the identity component of the diffeomorphism group of $M$ is meager whenever $M$ is a sphere or a closed connected 3-manifold. However the paper begins by proving some of the basic facts about bounded groups, and this part was already very interesting to me.

The groups $SL(n,R)$, $n\ge 2$, and $SL(n,Z)$, $n\ge 3$, are bounded. In the real case this follows from "a suitable version of the Gauss elimination process" to quote the paper... you have to show that any element of $SL(n,R)$ can be written as the product of a bounded number of conjugates of a bounded number of generators (namely, the elementary matrices with a 1 off the diagonal). Takes a moment's thought. Over the integers one needs also the bounded generation of $SL(n,Z)$ for $n\ge 3$ (Carter, David, and Gordon Keller. “Bounded Elementary Generation of SLn(O).” American Journal of Mathematics 105, no. 3 (June 1983): 673-687.)

An abelian group is bounded if and only if it is finite. This seems obvious, but there is no finite generation restriction here, so some care is needed in applying structure theory.

If $G$ surjects onto an unbounded group $H$, then $G$ is unbounded The basic idea is simple enough - take a conjugation-invariant norm on $H$ and pull it back via the surjection - but unfortunately the result need not be a norm (it vanishes on the whole kernel not just on the identity), so one needs to develop a more flexible theory of unbounded "quasi-norms" and prove that the existence of unbounded norms and quasi-norms are equivalent.

Wodzicki-Chern Classes

2010-05-27T10:42:00.001-04:00

So I took a look today at this new paper on the arXiv, by Steve Rosenberg and collaborators, that considers bundles (typically over infinite-dimensional manifolds) whose structural group is a subgroup of the invertible pseudodifferential operators of order $\le 0$ (on some other manifold).

A natural context for such a situation to arise is where your infinite-dimensional manifold is some (Sobolev-type completion $\mathcal M$ of) Maps(N,M), where N and M are ordinary finite dimensional closed manifolds - loop spaces being the canonical example of this kind of thing. Choosing metrics on N and M allows one to give the tangent bundle $T{\mathcal M}$, etc, the structure described above.

Now suppose we have a bundle with structure group $G\subseteq \Psi_{\le 0}$. We can try to use Chern-Weil theory to generate characteristic classes. In order to do this we need a trace on $\Psi_{\le 0}$ itself (just as standard Chern-Weil theory uses the trace on $M_n$ which is the Lie algebra of $GL(n)$).

The most natural example of such a trace is the famous Wodzicki residue which is in fact a trace on the algebra of all pseudodifferential operators. Inputting this trace into Chern-Weil theory yields characteristic classes called the Wodzicki-Chern classes.

The authors conjecture, and prove in some special cases, that these Wodzicki-Chern classes always vanish. (Of course this would mean that it is open season on secondary invariants related to these classes. The authors reference an earlier paper, "Riemannian geometry of loop spaces", where they discuss some nontrivial examples of Wodzicki-Chern-Simons classes.

In view of the importance of the Dixmier trace/Wodzicki residue in Connes' noncommutative geometry scheme, it would be interersting to know how this paper fits with the Connes picture.

A rigid framework

2010-05-26T09:51:00.001-04:00

Well, I am back from Yosemite, but not in quite the way I had hoped. I was climbing the Prow on Washington Column with Aaron McMillan (a grad student from Berkeley, student of Weinstein's) and on our second day I took a fall resulting in a broken ankle and the end of our climbing vacation. If you are interested in the long version you can find the story here.

Anyhow, I am sitting at home now with my leg encased in a rigid boot which will have to stay on for the next six weeks or so while the bones rejoin themselves. It got me thinking about the idea of such 'rigid frames' in teaching - actually in teaching analysis, since I'm thinking about my course for next fall. Bear with me for a moment while I try to explain what I mean.

Suppose that you're asked to give a proof of something like "the limit of a uniformly convergent sequence of continuous functions is continuous". As a professional mathematician you might just say "$3\epsilon$ argument", or you might write out a more detailed proof. But whatever you did - or, at least, whatever I would do - probably doesn't explicitly express every one of the many quantifiers that are involved in this statement, or explicitly delimit the scope of every one of the free variables that may be introduced during the proof. To be required to do so would be excessively rigid and constraining, like my ankle boot. As mathematicians we've developed the bones and muscles that allow us to work correctly with a less than wholly formal style of argumentation; and that's a vital skill. But, I'm wondering, do we give our students enough "rigid support" so that their mathematical "bones" can develop? Or do we overload them by presuming on strength that isn't there yet? If I just took off my boot now and tried to walk, the results would be disastrous; my bones aren't ready for that yet.

Specifically, one of the things that I try to emphasize in teaching analysis is taking apart an argument or definition involving multiple quantifiers into a hierarchy of more elementary units, which are nested within each other like subroutines in a computer program. And I then try to explain that to each of these elementary units corresponds a "proof skeleton", so that for instance to the elementary unit $\forall x\in A, P(x)$ ($P$ being some possibly complex proposition) corresponds the proof skeleton:

Let $x$ (or some other symbol not yet used) be an arbitrary member of $A$. Then (argument), leading to the conclusion $P(x)$. We have shown that $P(x)$ is truw for an arbitrary member $x\in A$, so we have proved $\forall x\in A, P(x)$. (end of scope of symbol $x$)

Nesting these proof skeleta in a way corresponding to the multiply-quantified statement to be proved gives a quite rigid framework - a "cast" - for the proof. Of course it is still necessary to supply the actual argument! In my experience though students sometimes need more guidance with the structure of the proof than the individual computations comprising it; and this system supplies it.

I am wondering about writing some software which will generate these "skeleta" semi-automatically and will force students to write proofs into them. Not for ever of course - just until the "bones" grow strong. Of course the worry is that then the teaching suddenly becomes about software and not about mathematics. Still, I think it could be a helpful tool. Maybe something like this exists already. Does anyone know?

Brief pause

2010-05-07T16:47:00.000-04:00

I won't be posting for a couple of weeks as I will be away climbing in Yosemite. I hope to get back to coarse remarks when I return :-)

Around soficity

2010-05-06T08:10:00.000-04:00

Andreas Thom just posted the article [1005.0823] About the metric approximation of Higman's group on the arXiv today. It is quite short with a specific result about Higman's group, but the introduction was most helpful to me in learning a bit about the ideas related to "soficity" of groups. It refers to another interesting paper: Elek, Gábor, and Endre Szabó. “Hyperlinearity, essentially free actions and L2-invariants. The sofic property.” Mathematische Annalen 332, no. 2 (4, 2005): 421-441.

It seems that these authors use some words like "hyperlinear" and "amenable action" in a sense different to that which is common to us in Baum-Connes land. for instance, for Elek-Szabo, the trivial action of a group on a point is *always* amenable.

Packing Tetrahedra

2010-05-04T09:23:00.000-04:00

This is some way from what this blog is supposed to be about, but like many packing problems it is fascinating and difficult. The question: How densely can regular tetrahedra be packed in 3-dimensional Euclidean space? Nobody knows, but here are some very interesting packings...

[1005.0011] Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

More about characterizations of exactness

2010-05-03T17:02:00.001-04:00

Following up an earlier post with some notes on the three papers below:

Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak [DN]
Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright [BNNW]
A note on topological amenability by Monod. [M]

All of these papers focus on the question of characterizing in "homological" terms what it is for a discrete group $G$ to be exact (or, more generally, to act amenably on some compact space --- it is known that exactness is equivalent to the amenability of the action of $G$ on its Stone-Cech compactification $\beta G$).

A necessary step along the way (again, in all the papers) is to relate the notion of exactness to some kind of "invariant mean". This follows a path explained by Johnson for ordinary amenability (Johnson, Barry Edward. 1972. Cohomology in Banach algebras. Providence, R.I.: American Mathematical Society.)

The paper [M] gives the greatest number of equivalent conditions (it is a legal requirement that all papers on amenability show that many conditions are equivalent). In particular let us consider the appropriate notion of "invariant mean". This is an element $\phi$ of the bidual $A^{**}$, where $A$ is the algebra of continuous functions on $\beta G$ with values in $\ell^1 G$ (also equivalent to the algebra of unconditionally convergent formal series $\sum_g f_g [g]$, with $f_g \in \ell^\infty(G)$); $\phi$ must be $G$-invariant and must sum to the constant function $1 \in \ell^\infty(G)$.

Aside: Monod also gives a couple of interesting alternative characterizations of $A$:

The space of compact operators on $\ell^1(G)$
The space of weak-$*$ - to - weak continuous operators on $\ell^\infty(G)$

The characterizations in [BNNW] seem quite close to that of [M]. Their invariant means are elements of a double dual $W_0^{**}$, where $W_0$ is defined a little differently but appears to be the subspace of $A$ consisting of elements that sum to a multiple of 1.

In [DN] the double dual of $W_0$ is taken in a more ambitious sense, as $hom(hom(W_0,C),C)$, where $C$ is the Banach space $\ell^\infty(G)$. But then one looks inside this double dual at the weak-$*$ closure, in an appropriate sense, of the members of $W_0$ itself. Now, let $R$ be some suitable ring of endomorphisms of the Banach space $C$ (e.g., the translation algebra). Then both $C$ and $hom(W_0,C)$ are $R$-modules and the elements of $hom(hom(W_0,C),C)$ coming from $W_0$ are $R$-module maps. Thus, it seems to me, one might as well restrict to the subspace of $R$-module maps from the start, and then much of the extra "size" of the double dual goes away. I think this may make a connection between the [DN] approach and the other two.

An update on random groups

2010-04-29T21:37:00.000-04:00

rgupdates.pdf application/pdf Object

technological toys

2010-04-29T08:59:00.000-04:00

Inspired by Nigel (who is often on the bleeding edge of technology) I ordered a Fujitsu ScanSnap - see below - which arrived a day ago. It's about the size of a loaf of bread, has one control button ("Scan"), and scans 20 double-sided pages per minute to PDF. I hope to use it to organize the piles of preprints, handwritten notes and manuscripts that I have accumulated in nearly thirty years as a mathematician.

That then begs the question - what software should I use to keep track of the resulting huge pile of PDFs? Right how I am working with Zotero which will organize pdfs, archive them on a WebDAV server, and also integrate with Penn State's library and other sources of bibliographic info. And it will seamlessly import the bibtex bibliography that I have maintained since I started using TeX. But there may well be other useful software packages out there that will do the same or better - any suggestions?

Various characterizations of exactness

2010-04-26T11:05:00.000-04:00

This post is a place to list a number of papers that have recently appeared on the arXiv which reformulate the notion of exactness for groups (or property A for spaces or amenable actions) in different ways:

A cohomological characterisation of Yu's Property A for metric spaces by Brodzki, Niblo and Wright.
Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak.
Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright.
A note on topological amenability by Monod.

I'll post more later about the relations between these.

Invariant translation approximation

2010-04-26T09:25:00.000-04:00

At the very end of my book Lectures on Coarse Geometry I asked the following question: suppose you take a discrete group $\Gamma$, consider it as a metric space and form the uniform translation algebra $UC^*(|\Gamma|)$. This algebra has a natural $\Gamma$-action and the $\Gamma$-fixed subalgebra, $UC^*(|\Gamma|)^\Gamma$, clearly contains the reduced $C^*$-algebra of the group $\Gamma$. Are these objects equal? In the book I showed that they are equal for amenable groups and outlined an argument, invented by Nigel Higson, which shows that they are also equal for free groups - this uses Haagerup's results about rapid decay.

It is clear that some kind of approximation property is involved here and in the book I called it the "invariant translation approximation property". (In our earlier discussions Nigel, Jerry and I were so irritated by this question that we called it the "completely stupid approximation property" but fortunately we were not completely stupid enough to use this term in print. Ooops...) Which groups possess this property?

While talking with Nowak in Texas I learned about a paper by Joachim Zacharias, On the invariant translation approximation property for discrete groups , which makes significant progress on this question. Zacharias' paper works as follows. First consider a strengthening of the ITAP by allowing coefficients: one looks at $UC^*(|\Gamma|;S)$ where $S$ is an auxiliary $C^*$-algebra (or operator space) and asks whether the $\Gamma$-invariant part of that is equal to $C^*_r(\Gamma)\otimes S$ (minimal tensor product). (N.B. There is no $\Gamma$-action on $S$ - no 'twisting'.) Zacharias proves that for exact groups this strengthened ITAP is equivalent to the Haagerup-Kraus approximation property ( Approximation properties for group $C^*$-algebras and group von Neumann algebras , Transactions of the American Mathematical Society, Vol. 344, No. 2 (Aug., 1994), pp. 667-699, which says that there is a net in the Fourier algebra $A(\Gamma)$ converging to 1 in a certain weak topology on the completely bounded multipliers on $C^*_r(\Gamma)$. Unfortunately no example of an exact discrete group without this property is known, but it has been conjectured that $SL(3,Z)$ is such a group.

On the way the author proves another characterization of exact groups, namely that $\Gamma$ is exact iff the map $S \mapsto UC^*(|\Gamma|;S)$ is an exact functor.

Geometry and complexity theory

2010-04-23T12:29:00.000-04:00

I'm at TAMU today, at the invitation of Piotr Nowak and Ron Douglas. Along with a number of others, they have made significant progress with understanding exactness of groups/property A in terms of appropriate notions of "invariant means" and "vanishing of bounded cohomology". I will probably write about this later.

However, while here I also had a chance to talk with Joseph Landsberg about his preprint P versus NP and geometry. Who could resist such a title? Here is my summary of what he told me. Suppose that you have to compute some huge polynomial in many variables. Then (obviously) in general it will take you a long time. As an example, consider the determinant of an $n\times n$ matrix, which is a polynomial of degree $n$ in $n^2$ variables. A brute force, term-by term evaluation takes a very long time. But in this case there is a trick - Gaussian elimination - which allows one to compute the polynomial much more quickly (with roughly $O(n^4)$ arithmetic operations I think). This is a hidden symmetry leading to speedy computation. Other polynomials, e.g. most famously the permanent (which is the same as the determinant but with all signs $+$), cannot (so far as is known) be computed in this easy way.

This leads to the notion of determinantal complexity (Valiant). You can envisage computing some polynomial such as the permanent of an $n\times n$ matrix by computing instead the determinant of some larger matrix built out of the original one in some way. (If the "larger matrix" is allowed to be sufficiently much larger one can always do this.) Define the determinantal complexity of the (sequence of) given polynomials to be the function that tells you how much you must increase the size of an $n\times n$ matrix to build a larger matrix that computes your polynomial via a determinat. Valiant conjectured that for the permanent, the determinantal compelxity grows faster than any polynomial. If I understand correctly, the falsity of this conjecture (i.e. a polynomial bound for $dc$ of the permanent) would imply $P=NP$.

To address this, the program of "geometric complexity theory" transfers the problem to one in algebraic or differential geometry. One looks at the variety defined by the determinant (in projective space) and allows the general (or special) linear group of the $n^2$ coordinates to act. The permanent (in some smaller degree $d$) defines a point in this space and the question becomes whether this point (or the Zariski closure of its orbit) lies in the Zariski closure of the orbit of the determinant. This question is then addressed either by representation theory (the ring of regular functions on a $G$-orbit can be completely described in terms of representation theory) or by local differential geometry.

I can write TeX!

2010-04-20T19:53:00.004-04:00

Seems as though I figured out how to include some TeX: $x^2+y^2+z^2=r^2$, $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. I got this from http://sumidiot.blogspot.com/2008/01/latex-in-blogger.html if you want to try it. You need to allow a javascript file to run from Nottingham University (UK).

The Atiyah conjecture is false (sort of)

2010-04-20T08:44:00.003-04:00

Diarmuid Crowley visited for a couple of days last week to talk about the Manifold Atlas Project which is a plan to produce a sort of online encyclopedia/journal of information about all sorts of manifolds. Of course we talked about other things also, and I learned from Diarmuid about a paper by Tim Austin of UCLA which gives a counterexample to a version of the "Atiyah Conjecture".

Atiyah would want me to point out that the conjecture is misnamed. In the paper in Asterisque where he introduces the L2 Betti numbers, Atiyah asked as a problem: "Give examples where these invariants are not integers or perhaps even irrational". The name "Atiyah conjecture" got attached to the claim that there are no such examples! And the question about the integrality of the invariants (for torsion free groups) is still open. But for groups with torsion, Zuk gave a counterexample some years ago to the claim that the denominators of the L2 Betti numbers must be generated by the torsion orders in the group; and now Austin shows that there are groups with irrational (even transcendental) L2 Betti numbers.

The argument is a non-constructive one: Austin builds an uncountable family of groups, and in the group ring of each group a particular element, such that the von Neumann dimensions of the kernels of these elements are all different. (The groups are certain "lamplighter-type" groups built on the free group.) Thus there are uncountably many different real numbers which are L2 Betti numbers, and some of them must not be rational (or algebraic). But the process doesn't identify a particular group for which this is true.

If I could figure out how to get TeX into this thing I might post more...

(Added later: See Thomas' comments below for a follow-up paper by Lukasz Grabowski, which begins "The main point of this article is to show some connections between Turing machines and von Neumann algebras".

Lin Shan's index theory

2010-04-20T08:32:00.001-04:00

OK, I am going to try to revive this blog in the hope that it will encourage me to read and keep up with the mathematical literature. We shall see...

Anyhow, I just wrote a review for Mathematical Reviews of the paper "Equivariant higher index theory and nonpositively curved manifolds" by Lin Shan (JFA 255(2008), 1480-1496. This paper defines and studies an analytic assembly map that includes both the coarse assembly map and the Baum-Connes assembly map as special cases, and it proves a Novikov conjecture type statement.

Infinite Expanders

2007-08-09T06:19:00.000-04:00

In a note published at

http://www.wisdom.weizmann.ac.il/~itai/infexp.ps

it is asked (by Binjamini I think), "Is there an infinite expander?".

By definition an infinite expander is an infinite connected bounded geometry graph with the following property: there exists a positive constant, call it c, such that for any set S of vertices (whether finite or not) and any ball B, less than half of whose points are in S, the ratio

(size of boundary S intersect B)/(size of S intersect B)

is greater than c.

The conjecture is that no such "infinite expander" exists.

QUESTIONS:

(a) What would it take for the graph of a group to be an infinite expander?
(b) Relate to the coarse property T problem.

2007-08-09T06:17:00.001-04:00

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

2006-06-07T07:43:00.000-04:00

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

This paper by Abreu-Suzuki gives a condition under which a coarse submersion between Riemannian manifolds is coarsely a product.

Interesting not only in itself but for its references - which are to another author (Kumeu) who came up with the basic coarse ideas, in the context of bg Riemannian manifolds, in the middle 1980s. I had not been aware of this before.

GT Monographs: Volume 9

2006-04-23T20:56:00.000-04:00

GT Monographs: Volume 9

This is the proceedings of an Oberwolfach conference on "exotic" homology manifolds. (Roughly speaking, these are manifolds for which the "zero'th Pontrjagin class" is not equal to 1.)

[math/0603675] The lower central series and pseudo-Anosov dilatations

2006-03-30T07:35:00.000-05:00

[math/0603675] The lower central series and pseudo-Anosov dilatations

The lower central series and pseudo-Anosov dilatations

Authors:
Benson Farb,
Christopher J. Leininger,
Dan Margalit

Comments: 26 pages, 6 figures

Subj-class: Geometric Topology; Dynamical Systems

MSC-class: 37E30 (Primary) 57M60, 37B40 (Secondary)

The theme of this paper is that algebraic complexity implies dynamical
complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g.
Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov
homeomorphism of S_g tends to zero at the rate 1/g. We consider here the
smallest dilatation of any pseudo-Anosov homeomorphism of S_g acting trivially
on Gamma/Gamma_k, the quotient of Gamma = pi_1(S_g) by the k-th term of its
lower central series, k > 0. In contrast to Penner's asymptotics, we prove that
this minimal dilatation is bounded above and below, independently of g, with
bounds tending to infinity with k. For example, in the case of the Torelli
group I(S_g), we prove that L(I(S_g)), the logarithm of the minimal dilatation
in I(S_g), satisfies .197 < L(I(S_g))< 4.127. In contrast, we find
pseudo-Anosov mapping classes acting trivially on Gamma/Gamma_k whose
asymptotic translation lengths on the complex of curves tend to 0 as g tends
toward infinity.

Full-text: PostScript, PDF, or Other formats

[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

2006-03-30T07:33:00.000-05:00

[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

All generating sets of all property T von Neumann algebras have free
entropy dimension $\leq 1$

Authors:
Kenley Jung,
Dimitri Shlyakhtenko

Comments: 6 pages

Subj-class: Operator Algebras

MSC-class: 46L54; 52C17

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an
arbitrary finite generating set of selfadjoint elements for N. By using
rigidity/deformation arguments applied to representations of N in full matrix
algebras, we deduce that the microstate spaces of X are asymptotically discrete
up to unitary conjugacy. We use this description to show that the free entropy
dimension of X, $\delta_0(X)$, is less than or equal to 1. It follows that when
N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then
$\delta_0(X)=1$ and otherwise, $\delta_0(X)=-\infinity$. This generalizes the
earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as
well as the results of Connes, Shlyakhtenko pertaining to group generators of
arbitrary property T algebras.

Full-text: PostScript, PDF, or Other formats

Which authors of this paper are endorsers?

[math/0603621] Property A, partial translation structures and uniform embeddings in groups

2006-03-28T06:37:00.000-05:00

[math/0603621] Property A, partial translation structures and uniform embeddings in groups

Brodzki, Niblo, Wright. - I blogged this before, but now there is an explicit invariant.

Interesting Mathematics

2006-03-03T16:12:00.000-05:00

Interesting Mathematics

Here is a paper by Ursula Hamenstadt proving exactness of the mapping class group. The method is interesting and is related to ideas of Kaimanovich. It is math.GR/0510116. One can look on her web page for other papers as well. http://www.math.uni-bonn.de/people/ursula/papers.html

[math/0603018] On the space of metrics with invertible Dirac operator

2006-03-02T09:24:00.000-05:00

[math/0603018] On the space of metrics with invertible Dirac operator

This paper by Mattias Dahl shows that several geometric constructions, eg codim-3 surgery, which one knows how to do in the category of positive scalar curvature manifolds, can in fact be done in the category of spin manifolds with invertible Dirac operator.

[math/0601700] Representations of residually finite groups by isometries of the Urysohn space

2006-01-31T06:27:00.000-05:00

[math/0601700] Representations of residually finite groups by isometries of the Urysohn space: "Representations of residually finite groups by isometries of the Urysohn space
Authors: Vladimir G. Pestov, Vladimir V. Uspenskij
Comments: 12 pages, LaTeX 2e
Subj-class: Representation Theory
MSC-class: 43A65; 20C99; 22A05; 22F05; 22F50; 54E50

As a consequence of Kirchberg's work, Connes Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the property (which we call Kirchberg's property) makes sense for an arbitrary topological group. We establish the validity of the Kirchberg property for the isometry group $Iso(U)$ of the universal Urysohn metric space $U$ as a consequence of a stronger result: every representation of a residually finite group by isometries of $U$ can be pointwise approximated by representations with a finite range. This brings up the natural question of which other concrete infinite-dimensional groups satisfy the Kirchberg property."

[math/0512592] Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

2006-01-26T05:56:00.000-05:00

[math/0512592] Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

Here's an interesting paper from the arXiv by Behrstock, Drutu and Mosher

[math/0302310] Hyperbolic group $C^$-algebras and free-product $C^$-algebras as compact quantum metric spaces

2005-12-31T17:34:00.000-05:00

[math/0302310] Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

An interesting paper on the metric geometry of the "dual absolute value of Dirac"

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

2005-12-02T06:23:00.000-05:00

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

From the arXiv

Authors: Tomasz Maszczyk,
Comments: 11 pages,
Subj-class: K-Theory and Homology; Mathematical Physics,
MSC-class: Primary 16E40, 17B35, 19K56, Secondary 46L87

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

Piotr Nowak's Homepage

2005-11-13T16:46:00.000-05:00

Piotr Nowak

I was going to reference just a couple of Piotr's articles but then I thought that I might as well point to his entire home page. Everything is good here! In particular he shows that coarse embeddability into a Hilbert space (or into ell-one) is not the same as property A. The example is devastatingly simple: take the disjoint union of n-fold products of copies of some finite group (e.g. the group of order 2). Notice that the spaces here are quasi-isometric to cubes in R^n with the ell-one metric. If one took the ell-two (Euclidean) metric instead, wouldn't one get Yu's old counterexample to coarse Baum-Connes? Something interesting seems to be going on here.

[Of course these aren't bg spaces. Is there a bg space with the same property?]

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

2005-11-07T05:59:00.000-05:00

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

Authors: Severino T. Melo, Cintia C. Silva
Subj-class: Operator Algebras; K-Theory and Homology
MSC-class: 46L80; 47G30

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

Another Coarse Geometry correction

2005-10-11T10:36:00.000-04:00

See this link for another correction to LCG, this time in the statement and proof of Rosenblatt's theorem in chapter 3. Thanks to Steve Ferrt and his students for spotting this one.

Best wishes

John

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

2005-09-28T20:24:00.000-04:00

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

We're familiar with the idea that statements about positive scalar curvature metrics (Gromov-Lawson-Rosenberg conjecture) and statements about higher signatures (Novikov conjecture) have certain parallels - ultimately they involve the higher index theorem applied to different elliptic operators, Dirac in the first case and signature in the second.

In this new paper Jonathan Rosenberg proposes a further family of statements involving higher index theory for the Dolbeault operator. These are statements in complex algebraic geometry about "higher Todd genera" for varieties.

This could be a whole new playground for higher index theorists.

Discrete Morse theory and graph braid groups

2005-08-31T20:18:00.000-04:00

AGT 5 (2005) Paper 44 (Abstract)

Discrete Morse theory and graph braid groups
Daniel Farley, Lucas Sabalka
Abstract. If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

Subelliptic spinc Dirac operators

2005-08-31T14:36:00.000-04:00

Charlie Epstein kindly gave me permission to publish this paper on index theory for generalizations of the d-bar Neumann problem and its relationship to index theory for Fourier integral operators.

[math/0505622] Free construction of CAT(1) spaces

2005-08-30T07:07:00.000-04:00

[math/0505622] Free construction of CAT(1) spaces

[math/0508135] On the generalized Nielsen realization problem

2005-08-09T07:21:00.000-04:00

[math/0508135] On the generalized Nielsen realization problem

Authors: Jonathan Block, Shmuel Weinberger
Subj-class: Geometric Topology
MSC-class: 57N

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

GT Vol 9 (2005) Paper 34 Hadamard spaces with isolated flats

2005-08-08T21:18:00.000-04:00

GT Vol 9 (2005) Paper 34 (Abstract)

By Hruska and Kleiner with Hindawi

[math/0507542] A new kind of index theorem

2005-07-27T07:15:00.000-04:00

[math/0507542] A new kind of index theorem

This is Ron Douglas' talk at the Kaminker retirement conference.

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations

2005-07-26T06:12:00.000-04:00

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations

Ruan's lectures on orbifolds

2005-07-21T10:52:00.000-04:00

Ruan's lectures at MSRI

[math/0502388] Quotients of Standard Hilbert Modules

2005-07-18T06:17:00.000-04:00

[math/0502388] Quotients of Standard Hilbert Modules

Kind of a synthesis of BDF and algebraic geometry. By Bill Arveson.

A Comparison of Leibniz and Cyclic Homologies

2005-07-11T06:43:00.000-04:00

Authors: Jerry Lodder
Comments: 14 pages
Subj-class: K-Theory and Homology; Algebraic Topology
MSC-class: 19D55; 17B66; 17A32

We relate Leibniz homology to cyclic homology by studying a map from a long exact sequence in the Leibniz theory to the ISB periodicity sequence in the cyclic theory. This provides a setting by which the two theories can be compared via the 5-lemma. We then show that the Godbillon-Vey invariant, as detected by the Leibniz homology of formal vector fields, maps to the Godbillon-Vey invariant as detected by the cyclic homology of the universal enveloping algebra of these vector fields.

2005-07-08T07:27:00.000-04:00

Geometry of K\"ahler Metrics and Foliations by Holomorphic Discs

I've always been interested in holomorphic foliations...

[math/0506547] Coarse dimensions and partitions of unity

2005-06-29T12:03:00.000-04:00

[math/0506547] Coarse dimensions and partitions of unity

More development of "coarse dimension theory" by professional dimension-theorists.

a space that can't embed in any uniformly convex Banach space

2005-06-28T10:29:00.000-04:00

http://arxiv.org/abs/math/0506178

[math/0506544] Isometries, rigidity, and universal covers

2005-06-28T10:28:00.000-04:00

[math/0506544] Isometries, rigidity, and universal covers

Farb and Weinberger classify aspherical Riemannian manifolds whose universal cover has "extra symmetry"

Coarse Homotopy - Trung's Thesis

2005-06-21T09:31:00.000-04:00

This is a small advertising memo about part of Viet-Trung Luu's forthcoming thesis, which contains a very elegant unified treatment of various theorems about the 'coarse homotopy' or 'Lipschitz homotopy' invariance of the K-theory of C*(X). It formalizes the basic idea of the paper by Nigel and me in the Transactions, found here. One needs "C*(X) with coefficients", defined using Hilbert modules. Trung's approach has a geometric and an analytic part. The geometric part is to show that a coarse homotopy gives an element of C*(X; C[0,1]). The analytic part is to show that there is a natural pairing between the K-theory of C*(X;D) and the K-homology of D, with values in the K-theory of C*(X). Now one uses the known homotopy invariance of K-homology - the fact that the inclusions of 0 and 1 into [0,1] induce isomorphisms on K-homology - to conclude the coarse homotopy invariance of K_*(C*(X)).

[math/0506361] Property (T) and rigidity for actions on Banach spaces

2005-06-21T07:30:00.000-04:00

[math/0506361] Property (T) and rigidity for actions on Banach spaces

Here the authors study variations of property T for actions on Banach spaces, especially Lp. When p is not 2 the equivalence between the original version of property T (isolation of the trivial representation) and the fixed point version F (affine isometric actions) does not hold.
They prove that every property T group has T(L^p) for all p, and F(L^p) for p<2+\epsilon.

Corrections to "Lectures on Coarse Geometry"

2005-06-17T14:45:00.000-04:00

Corrections to "Lectures on Coarse Geometry"

Bernd Grave pointed out a couple of errors in Lectures on Coarse Geometry, explained here.

One is a foolish mistake I made with Urysohn's lemma. Given two disjoint closed subsets of a compact Hausdorff space, Urysohn's lemma allows one to find a continuous function equal to zero on one set, one on the other. But in general the set of points where said function is zero will be *larger* than the originally given closed set. Only in metric spaces can one expect to arrange equality.

Property A and nuclearity of Roe algebras (application/pdf Object)

2005-06-15T08:48:00.000-04:00

propAnuclear.pdf (application/pdf Object)

By Brodzki, Niblo and Wright. One knows that for a discrete group G, property A is equivalent to exactness/nuclearity of its translation algebra. But the same thing is not known for a general (uniformly discrete) coarse space. In this paper is a measure of the 'grouplikeness' of a coarse space in terms of "partial translation structures". If X is uniformly embeddable in a discrete group, then it has a "free partial translation structure" and the expected equivalence holds.

Comments: Gromov's counterexample construction gives a way of building an 'almost' injective embedding of a more-or-less arbitrary coarse space into a discrete group. Relate that to this paper!

[math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

2005-06-15T07:32:00.000-04:00

[math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

wall.pdf (application/pdf Object)

2005-06-14T21:45:00.000-04:00

wall.pdf (application/pdf Object)

Cambridge lecture notes on differential topology (ca. 1960) by C.T.C. Wall.

Detailed treatment of foundational theorems.

hgw.pdf (application/pdf Object)

2005-06-14T21:25:00.000-04:00

hgw.pdf (application/pdf Object)

Guentner, Higson and Weinberger prove the Novikov conjecture for all linear groups. The proof goes via coarse embeddings into Hilbert space.

cime.pdf (application/pdf Object)

2005-06-14T21:23:00.000-04:00

cime.pdf (application/pdf Object)

Nigel Higson's notes on C*-algebras, K-theory and the Baum-Connes conjecture. Includes discussion of the "spectral picture" of K-theory; the E-theory approach to the Baum Connes conjecture; some counter-example stuff.

Mendel-Naor - cotype

2005-06-14T15:59:00.001-04:00

Metric Cotype

[This gives the definitive answer to the question when Lp is uniformly (coarsely) embeddable into Lq.]

Title: Metric Cotype
Authors: Manor Mendel, Assaf Naor
Comments: 42 pages
Subj-class: Functional Analysis; Metric Geometry
MSC-class: 46B20; 51F99
\\
We introduce the notion of cotype of a metric space, and prove that
for Banach spaces it coincides with the classical notion of Rademacher
cotype. This yields a concrete version of Ribe's theorem, settling a
long standing open problem in the non-linear theory of Banach spaces.
We apply our results to several problems in metric geometry. Namely,
we use metric cotype in the study of uniform and coarse embeddings,
settling in particular the problem of classifying when L_p coarsely or
uniformly embeds into L_q. We also prove a non-linear analog of the
Maurey-Pisier theorem, and use it to answer a question posed by Arora,
Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain
quantitative bounds in a metric Ramsey theorem due to Matousek.

Pseudocharacters

2005-06-14T15:56:00.000-04:00

Geometry and Topology, Volume 9 (2005) Paper no. 26, pages 1147--1185

Geometry of pseudocharacters

Author(s):
Jason Fox Manning

Abstract:

If G is a group, a pseudocharacter f: G-->R is a function which is
"almost" a homomorphism. [in a coarse-geometric sense: JR]
If G admits a nontrivial pseudocharacter f,
we define the space of ends of G relative to f and show that if the
space of ends is complicated enough, then G contains a nonabelian free
group. We also construct a quasi-action by G on a tree whose space of
ends contains the space of ends of G relative to f. This construction
gives rise to examples of "exotic" quasi-actions on trees.