Thompson groups

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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Group (mathematics) Algebraic structure with one binary operation

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980.

Contents

The Thompson groups, and F in particular, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.

Simple group Group without normal subgroups other than the trivial group and itself

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

In mathematics, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.

Subgroup Subset of a group that forms a group itself

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted HG, read as "H is a subgroup of G".

It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".

Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.

Presentations

A finite presentation of F is given by the following expression:

where [x,y] is the usual group theory commutator, [{text|xyx}}−1y−1.

Although F has a finite presentation with 2 generators and 2 relations, it is most easily and intuitively described by the infinite presentation:

The two presentations are related by x0=A, xn = A1nBAn1 for n>0.

Other representations

The Thompson group F is generated by operations like this on binary trees. Here L and T are nodes, but A B and R can be replaced by more general trees. AA Tree Skew2.svg
The Thompson group F is generated by operations like this on binary trees. Here L and T are nodes, but AB and R can be replaced by more general trees.

The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2.

Binary tree tree data structure in which each node has at most two children

In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set. Some authors allow the binary tree to be the empty set as well.

Homeomorphism In mathematics, isomorphism of topological spaces

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I . In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.

The group F can also be considered as acting on the unit circle by identifying the two endpoints of the unit interval, and the group T is then the group of automorphisms of the unit circle obtained by adding the homeomorphism xx+1/2 mod 1 to F. On binary trees this corresponds to exchanging the two trees below the root. The group V is obtained from T by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).

The Thompson group F is the group of order-preserving automorphisms of the free Jónsson–Tarski algebra on one generator.

Amenability

The conjecture of Thompson that F is not amenable was further popularized by R. Geoghegan --- see also the Cannon-Floyd-Parry article cited in the references below. Its current status is open: E. Shavgulidze [1] published a paper in 2009 in which he claimed to prove that F is amenable, but an error was found, as is explained in the MR review.

It is known that F is not elementary amenable, see Theorem 4.10 in Cannon-Floyd-Parry. If F is not amenable, then it would be another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups, which suggested that a finitely-presented group is amenable if and only if it does not contain a copy of the free group of rank 2.

Connections with topology

The group F was rediscovered at least twice by topologists during the 1970s. In a paper which was only published much later but was in circulation as a preprint at that time, P. Freyd and A. Heller [2] showed that the shift map on F induces an unsplittable homotopy idempotent on the Eilenberg-MacLane space K(F,1) and that this is universal in an interesting sense. This is explained in detail in Geoghegan's book (see references below). Independently, J. Dydak and P. Minc [3] created a less well-known model of F in connection with a problem in shape theory.

In 1979, R. Geoghegan made four conjectures about F: (1) F has type FP; (2) All homotopy groups of F at infinity are trivial; (3) F has no non-abelian free subgroups; (4) F is non-amenable. (1) was proved by K. S. Brown and R. Geoghegan in a strong form: there is a K(F,1) with two cells in each positive dimension. [4] (2) was also proved by Brown and Geoghegan [5] in the sense that the cohomology H*(F,ZF) was shown to be trivial; since a previous theorem of M. Mihalik [6] implies that F is simply connected at infinity, and the stated result implies that all homology at infinity vanishes, the claim about homotopy groups follows. (3) was proved by M. Brin and C. Squier. [7] The status of (4) is discussed above.

It is unknown if F satisfies the Farrell–Jones conjecture. It is even unknown if the Whitehead group of F (see Whitehead torsion) or the projective class group of F (see Wall's finiteness obstruction) is trivial, though it easily shown that F satisfies the Strong Bass Conjecture.

D. Farley [8] has shown that F acts as deck transformations on a locally finite CAT(0) cubical complex (necessarily of infinite dimension). A consequence is that F satisfies the Baum-Connes conjecture.

See also

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References

  1. Shavgulidze, E. (2009), "The Thompson group F is amenable", Infinite Dimensional Analysis, Quantum Probability and Related Topics, 12 (2): 173–191, doi:10.1142/s0219025709003719, MR   2541392
  2. Freyd, Peter; Heller, Alex (1993), "Splitting homotopy idempotents", Journal of Pure and Applied Algebra, 89 (1–2): 93–106, doi:10.1016/0022-4049(93)90088-b, MR   1239554
  3. Dydak, Jerzy; Minc, Piotr (1977), "A simple proof that pointed FANR-spaces are regular fundamental retracts of ANR's", Bulletin de l'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, 25: 55–62, MR   0442918
  4. Brown, K.S.; Geoghegan, Ross (1984), An infinite-dimensional torsion-free FP_infinity group, 77, pp. 367–381, Bibcode:1984InMat..77..367B, doi:10.1007/bf01388451, MR   0752825
  5. Brown, K.S.; Geoghegan, Ross (1985), "Cohomology with free coefficients of the fundamental group of a graph of groups", Commentarii Mathematici Helvetici, 60: 31–45, doi:10.1007/bf02567398, MR   0787660
  6. Mihalik, M. (1985), "Ends of groups with the integers as quotient", Journal of Pure and Applied Algebra, 35: 305–320, doi:10.1016/0022-4049(85)90048-9, MR   0777262
  7. Brin, Matthew.; Squier, Craig (1985), "Groups of piecewise linear homeomorphisms of the real line", Inventiones Mathematicae, 79 (3): 485–498, Bibcode:1985InMat..79..485B, doi:10.1007/bf01388519, MR   0782231
  8. Farley, D. (2003), "Finiteness and CAT(0) properties of diagram groups", Topology, 42 (5): 1065–1082, doi:10.1016/s0040-9383(02)00029-0, MR   1978047