Bounded reduction of invertible matrices over polynomial rings by addition operations

Kazhdan [15] introduced a property T for topological groups and proved it for the discrete groups SLrZ with n ≥ 3. Here Z stands for the ring of integers. He also proved that the group SL2Z did not enjoy the T -property. He believed that neither SL3(Z[x]) did. The T -property was related with other group properties (cf., e.g., [28]). For example, here are two properties of G = SLrZ, r ≥ 3, which G = SL2Z does not have: every matrix in G is a product of a bounded number of elementary matrices [8]; every subgroup of a finite index in G contains a congruence subgroup and hence has no infinite commutative factor groups [25], [3]. Shalom [29] (cf. also [10]) proved that if every matrix in the group

[1]  H. Park,et al.  An Algorithmic Proof of Suslin′s Stability Theorem for Polynomial Rings , 1994, alg-geom/9405003.

[2]  J. Gago-Vargas.  Constructions in R[x1,…,xn]: applications to K-theory , 2002 .

[3]  Hyman Bass,et al.  Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2) , 1967 .

[4]  Dante C. Youla,et al.  The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices , 1984 .

[5]  Nirmal K. Bose,et al.  Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets , 2004 .

[6]  Hyman Bass,et al.  K-Theory and stable algebra , 1964 .

[7]  F. Grunewald,et al.  On symplectic groups over polynomial rings , 1991 .

[8]  R. Laubenbacher,et al.  A new algorithm for the Quillen-Suslin theorem. , 2000 .

[9]  N. Bose,et al.  A Generalization of Serre's Conjecture and Some Related Issues , 2001 .

[10]  Zhiping Lin,et al.  Notes on n-D Polynomial Matrix Factorizations , 1999, Multidimens. Syst. Signal Process..

[11]  P. M. Cohn,et al.  On the structure of the GL2 of a ring , 1966 .

[12]  Hyungju Park.  A Realization Algorithm for SL2(R[x1, ... , xm]) over the Euclidean Domain , 1999, SIAM J. Matrix Anal. Appl..

[13]  Joos Heintz,et al.  Algorithmic aspects of Suslin's proof of Serre's conjecture , 2005, computational complexity.

[14]  J. Mennicke.  Finite Factor Groups of the Unimodular Group , 1965 .

[15]  A. C. Tan,et al.  On a generalized factorization problem for structurally passive synthesis of digital filters , 1989 .

[16]  Henri Lombardi,et al.  Suslin's algorithms for reduction of unimodular rows , 2005, J. Symb. Comput..

[17]  Zhiping Lin,et al.  FACTORIZATIONS FOR nD POLYNOMIAL MATRICES* , 2005 .

[18]  B. Sturmfels,et al.  Algorithms for the Quillen-Suslin theorem , 1992 .

[19]  Yehuda Shalom,et al.  Bounded generation and Kazhdan’s property (T) , 1999 .

[20]  Yves Cornulier.  PROPERTY T FOR LINEAR GROUPS OVER RINGS, AFTER SHALOM , 2007 .

[21]  Friedrich Hirzebruch,et al.  On Artin's Conjecture and Euclid's Algorithm in Global Fields , 1977 .

[22]  Andrei Suslin,et al.  ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS , 1977 .

[23]  Ettore Fornasini,et al.  nD Polynomial Matrices with Applications to Multidimensional Signal Analysis , 1997, Multidimens. Syst. Signal Process..

[24]  W. V. D. Kallen.  SL3 (C[X]) does not have bounded word length , 1982 .