Units of Integral Group Rings, Banff, 2014 Eric Jespers Vrije Universiteit Brussel Belgium
G a finite group ZG integral group ring U(ZG) group of invertible elements, a finitely presented group PROBLEM (Sehgal-Ritter, 1989): describe constructively finitely many generators, in a generic way PROBLEM (Sehgal): describe algebraic structure of U(ZG) PROBLEM: Construct units
ZG QG = i M ni (D i ), each D i a Q-finite dimensional division algebra. ZG O = i ZGe i QG containment of (Z-)orders (finitely generated Z-modules that contain a Q-basis of QG) e i primitive central idempotents unit groups of orders are commensurable If O is an order in a finite dimensional semisimple rational algebra A then O 1, U(Z(O)) has finite index in U(O). If, moreover, A is simple then O 1 U(Z(O)) is cyclic and finite. may take O = M n1 (O 1 ) M n2 (O 2 ) M n (O n ), O 1 reduced norm one elements, O i order in D i
Solve problem first for central units Solve problem in each SL ni (O i ) Make constructions such that generators live in ZG, or pull back into ZG
Bass units u k,m (g) = (1 + g + g 2 + + g k 1 ) m + 1 km g, g where g G and k and m are positive integers such that k m = 1 mod g.
Bass units u k,m (g) = (1 + g + g 2 + + g k 1 ) m + 1 km g, g where g G and k and m are positive integers such that k m = 1 mod g. Related to cyclotomic units 1 ζ k 1 ζ = 1 + ζ + + ζk 1
Central Units G abelian: specific set of u k,mk,c (a C ) that is a basis for a subgroup of finite index in U(ZG) Bass-Milnor, recent construction by JRV, without use of K-theory, main issue: construct units that only contribute to one simple factor G not abelian: for some classes of G (abelian-by-supersolvable) construction possible that is a product of conjugates of Bass units and that yields a basis for a subgroup of finite index in Z(U(ZG)). JORV 2013 possible: because one gets a method to convert some units into central units (products of conjugates)
Let g G of order not a divisor of 4 or 6 and let N : N 0 = g N 1 N 2 N m = G be a subnormal series in G. For u U(Z g ) define c0 N (u) = u and c N i (u) = ci 1(u) N h, h T i where T i is a transversal for N i 1 in N i, i 1.
Bicyclic units g G g = g 1 i=0 Ritter-Sehgal introduced the bicyclic units g i b(g, h) = 1 + (1 h)g h and b( h, g) = 1 + hg(1 h) (g, h G),
Bicyclic units g G g = g 1 i=0 Ritter-Sehgal introduced the bicyclic units g i b(g, h) = 1 + (1 h)g h and b( h, g) = 1 + hg(1 h) (g, h G), Related to 1 + (1 e)αe and 1 + eα(1 e), with e 2 = e,... elementary matrices
Theorem (Jespers-Leal, Ritter-Sehgal) Suppose G does not have a non-abelian epimorphic image that is fixed point free and QG does not have exceptional simple images. Then group generated by Bass units and bicyclic units is of finite index in U(ZG).
Definition A simple finite dimensional rational algebra is said to be exceptional if it is one of the following types: 1. a non-comm. div. algebra (not totally definite quat. alg.), 2. M 2 (Q), 3. M 2 (F ) with F a quadratic imaginary extension of Q, (( )) 4. M a,b 2 Q, with a and b negative integers (i.e. totally definite quaternion algebra with center Q). ( a,b Q ) is a previous talk gives algorithms recent other algorithms via actions of discontinuous groups on hyperbolic spaces (JKR) case (4) fully described now, reduces to orders that are Euclidean generic constructions in orders in division algebras remain a black box
structure theorem A rational algebra A is said to be of Kleinian type if it is either a number field, or a quaternion algebra over a number field F such that σ(o 1 ) is a discrete subgroup of SL 2 (C) for some embedding σ of F in C and some order O of A. Theorem (JPRRZ 2007) Let A = k i=1 A i with each A i a finite dimensional semisimple rational algebra. Let O be an order in A and for each A i let O i be an order in A i. Then U(O) is virtually a direct product of free-by-free groups (respectively, of free groups) if and only if each Oi 1 is virtually free-by-free (respectively, virtually free).
Theorem (JPRRZ 2007) For a finite group G the following statements are equivalent. (A) U(ZG) is virtually a direct product of free-by-free groups. (B) For every simple quotient A of QG and some (every) order O in A, O 1 is virtually free-by-free. (D) G is of Kleinian type, i.e. QG is an algebra of Kleinian type. (E) Every simple quotient of QG is either a field, a totally definite quaternion algebra or M 2 (K), where K is either Q, Q(i), Q( 2) or Q( 3). (F) G is either abelian or an epimorphic image of A H, where A is abelian group of exponent 2,4 or 6 and H can in each case be well described.
Problem: describe units in orders in finite dimensional rational division algebras