Serre's property FA

Last updated March 18, 2019

In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a group is a set equipped with a binary operation which 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.

Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003.

A group G is said to have property FA if every action of G on a tree has a global fixed point.

In mathematics, a group action is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

In mathematics, and, more specifically, in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. Every acyclic connected graph is a tree, and vice versa. A forest is a disjoint union of trees, or equivalently an acyclic graph that is not necessarily connected.

In mathematics, a fixed point of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f(x) if f(c) = c. This means f(f ) = fⁿ(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.

Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.

In mathematics, the HNN extension is an important construction of combinatorial group theory.

Generating set of a group Subset of a group such that all group elements can be expressed by finitely many group operations on its elements

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their inverses.

Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup.

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number.

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.

Examples of groups with property FA include SL₃(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL₂(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C₄ and C₆ along C₂.

Any quotient group of a group with property FA has property FA. If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general. If N is a normal subgroup of G and both N and G/N have property FA, then so does G.

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G : H| or [G : H] or (G:H).

In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $H$ of a group $G$ is normal in $G$ if and only if $gH = Hg$ for all $g$ in $G$ . The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup $H$ of a group $G$ is normal in $G$ . Normal subgroups can be used to construct quotient groups from a given group.

It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.

Examples

The following groups have property FA:

A finitely generated torsion group;
SL₃(Z);
The Schwarz group $\left\langle {a,b:a^{A}=b^{B}=(ab)^{C}=1}\right\rangle$ for integers A,B,C ≥ 2;
SL₂(R) where R is the ring of integers of an algebraic number field which is not Q or an imaginary quadratic field.

The following groups do not have property FA:

SL₂(Z);
SL₂(R_D) where R_D is the ring of integers of an imaginary quadratic field of discriminant not −3 or −4.

Related Research Articles

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups.

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators. We then say G has presentation

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, such that the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example $They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.$

In number theory and algebraic geometry, a modular curveY(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curvesX(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.

In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. The formal definition, introduced by David Kazhdan (1967), gives this a precise, quantitative meaning.

In mathematics, a pro-p group is a profinite group $such that for any open normal subgroup the quotient group is a p -group . Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite.$

In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.

In group theory, a branch of abstract algebra, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p¹⁺²ⁿ. Extra special groups often occur in centralizers of involutions. The ordinary character theory of extra special groups is well understood.

In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem.

Lattice (discrete subgroup) discrete subgroup in a locally compact topological group

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rⁿ, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.

Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.

In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russian mathematician, in 1934. Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of its intersections with the conjugates of the factors of the original free product.

In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.

In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group.

In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two finitely generated subgroups of this group is again finitely generated. The property is named after Albert G. Howson who in a 1954 paper established that free groups have this property.

References

Serre, Jean-Pierre (1974). "Amalgames et points fixes". Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics (in French). 372. pp. 633–640. MR 0376882. Zbl 0308.20026.
Serre, Jean-Pierre (1977). Arbres, amalgames, SL₂. Astérisque (in French). 46. Société Mathématique de France. Zbl 0369.20013. English translation: Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.
Watatani, Yasuo (1981). "Property T of Kazhdan implies property FA of Serre". Math. Japon. 27: 97–103. MR 0649023. Zbl 0489.20022.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.