All braid groups are linear

You need to know: Matrix multiplication, invertible matrix, group, subgroup, group isomorphism, field, braid group.

Background: General linear group of degree n over field F, denoted as GL(n, F), is the set of $n \times n$ invertible matrices with entries from F and with matrix multiplication as the group operation. A group is said to be linear if it is isomorphic to a subgroup of GL(n, F) for some natural number n and some field F.

The Theorem: On 11th March 2000 Daan Krammer submitted to Annals of Mathematics a paper in which he proved that all braid groups are linear.

Short context: In 1935, Burau suggested a way to represent elements of any braid group (braids) as matrices over some field. However, he did not prove whether it is faithful (that is, whether different braids corresponds to different matrices). In 1991, Moody showed that this not always the case. The Theorem proves that a different representation, suggested by Krammer in 1999, is faithful, thus establishing that all braid groups are linear. In May 2000, the same result was proved independently by Bigelow.

Links: Free arxiv version of the Bigelow’s paper is here, journal version of Krammer’s paper is here. See also Section 2.2 of this book for an accessible description of the Theorem.

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There exists a field with u-invariant 9

You need to know: The concept of a Field.

Background: A number u(F) is called u-invariant of a field F, if (a) equation $\sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j = 0$ (where $x_1, x_2, \dots, x_n$ are variables and $a_{ij}$ are some coefficients from F) in n=u(F) variables may have no solutions, except of $x_1 = x_2 = \dots = x_n = 0$ , and (b) the same equation in $n=u(F)+1$ variables always have a non-zero solution.

The Theorem: On 4th January 2000 Oleg Izhboldin submitted to Annals of Mathematics a paper in which he proved that there exists a field F with u-invariant 9.

Short context: In 1953, Kaplansky conjectured that u-invariant of any field is always a power of 2. In 1989, Merkurev disproved this, and showed that u-invariant can be any even number, but left open whether it can be any odd number grater than 1. It is known that u-invariant can never be 3, 5, or 7. However, the Theorem proves that it can be 9.

Links: The original paper is available here. See also Section 1.7 of this book for an accessible description of the Theorem.

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Elements of finite simple groups have short normal subsets representations

You need to know: Group, finite group, subgroup, identity element e of a group, logarithm $\log$ .

Background: A subset S of group G is called normal, if $g\cdot a\cdot g^{-1}\in S$ for any $a \in S$ and $g \in G$ . A group G is called simple if it does not have any normal subgroup, except of itself and $\{e\}$ , where $\{e\}$ is the set consisting on identity element only. For any finite set T, denote $|T|$ the number of elements in it.

The Theorem: On 28th October 1999, Martin Liebeck and Aner Shalev submitted to Annals of Mathematics a paper in which they proved, among other results, that there exists a constant c such that if G is a finite simple group and $S\neq \{e\}$ is a normal subset of G, then, for any $m \geq c \log |G|/\log |S|$ , any element of G can be expressed as a product of m elements of S.

Short context: Because there are $|S|^m$ ways to write a product of m elements of S, the conclusion in the Theorem cannot hold if $|S|^m < |G|$ , or $m< \log |G|/\log |S|$ . Hence, the inequality $m \geq c \log |G|/\log |S|$ in the Theorem is the best possible up to constant factor. The Theorem has many applications, see here for an example.

Links: The original paper is here. See also Section 1.6 of this book for an accessible description of the Theorem.

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There exist two different finite groups with isomorphic integral group rings

You need to know: Groups, finite groups, rings, isomorphism for groups and rings.

Background: For any (finite) group G, an integral group ring is the ring whose elements are all possible sums $\sum\limits_{g\in G}a_gg$ , where $a_g$ are integers, addition is defined as $\sum_{g\in G}a_gg + \sum_{g\in G}b_gg = \sum_{g\in G}(a_g+b_g)g,$ and multiplication is given by $(\sum_{g\in G}a_gg)\cdot(\sum_{h\in G}b_hh) = \sum_{g\in G}\sum_{h\in G}a_gb_hgh$ .

The Theorem: On 15th September 1999, Martin Hertweck submitted to Annals of Mathematics a paper in which he proved that there exist two finite non-isomorphic groups G and H such that their integral group rings are isomorphic.

Short context: Obviously, if two groups G and H are isomorphic, then so are their integral group rings. In 1940, Higman asked if the converse is true, that is, whether different finite groups have different integral group rings. The question was answered positively in many special cases. The Theorem, however, implies that in general the answer is negative.

Links: The original paper is here. See also Section 1.5 of this book for an accessible description of the Theorem.

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Characterisation of ﬁnitely generated groups with word problem in NP.

You need to know: Class P, class NP, NP-complete problems, group, presentation of a group in the form $<S|R>$ where S is a set of generators and R is a set of relations.

Background: A group G is called finitely generated if it has a presentation $<S|R>$ with finite number of generators. If the set of relations is finite as well, G is called finitely presented. A word in group G is any product of generators and their inverse elements. The word problem in G is the problem to decide whether a given word represents an identity element.

For any word a representing the identity element e, there is a sequence of steps that reduces a to the empty word, where each step is a free reduction or insertion or the application of a relator. We call the number of applications of relators in a sequence its cost, and let T(a) be the minimum cost of a sequence that transforms a into e. A function f is called an isoperimetric function of G if $T(a) \leq f(|a|)$ for any a representing e, where $|a|$ is the length of word a. We say that isoperimetric function f is polynomial, if $f(n) = cn^k$ for some constants c and k.

The Theorem: On 20th April 1998, Jean-Camille Birget, Alexander Olshanskii, Eliyahu Rips and Mark Sapir submitted to Annals of Mathematics a paper in which they proved that the word problem of a ﬁnitely generated group H is in NP if and only if H is a subgroup of a ﬁnitely presented group G with a polynomial isoperimetric function.

Short context: Every group H with polynomial isoperimetric function is known to be in NP. The converse is not true: there are groups H with word problem in NP but such that every isoperimetric function grows very fast. The Theorem, however, shows that every such group H can be embedded into a different group G which is finitely presented and has a polynomial isoperimetric function.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.9 of this book for an accessible description of the Theorem.

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A finitely presented group can have NP-complete word problem.

You need to know: Class P, class NP, NP-complete problems, group, presentation of a group in the form $<S|R>$ , where S is a set of generators and R is a set of relations.

Background: A group G is called finitely presented if it has a presentation $<S|R>$ with finite number of generators and finite set of relations. A word in group G is any product of generators and their inverse elements. The word problem in G is the problem to decide whether a given word represents the identity element.

The Theorem: On 24th May 1997, Mark Sapir, Jean-Camille Birget, and Eliyahu Rips submitted to the Annals of Mathematics a paper in which they proved, among other results, that there exists a finitely presented group for which the word problem is NP-complete.

Short context: In general, the word problem for a finitely presented group may be well outside of NP. In fact, Novikov proved in 1955 that there are finitely presented groups for which the word problem is undecidable. The Theorem constructs the first example of a finitely presented group whose word problem is in NP but, still, it is unlikely that there is an efficient algorithm for solving it.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.8 of this book for an accessible description of the Theorem.

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