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Register a new userThe Auslander conjecture for dimension less then 7
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In 1964 L. Auslander conjectured that every crystallographic subgroup of an the affine group is virtually solvable, i.e. contains a solvable subgroup of finite index. D. Fried and W. Goldman proved Auslanders conjecture for n = 3 using cohomological arguments. We prove the Auslander conjecture for n < 7. The proof is based mainly on dynamical arguments. In some cases, we use the cohomological argument which we could avoid but it would significantly lengthen the proof.
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For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${rm Hom}_R(M,R)$ and ${rm Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.
In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property FA_n, an analogue of Serres property FA for actions on CAT(0) complexes. Property FA_n has implications for irreducible representations and complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies FA_n and show that this condition is in fact equivalent to FA_n for n=1 and 2. As part of the proof, we compute the Gersten-Stallings angles between special subgroups of Coxeter groups.
We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that begin{equation*} sum_{g in U backslash G/W} bar d(U cap gWg^{-1}) leq bar d(U) bar d(W) end{equation*} where $bar d(K) = max{d(K) - 1, 0}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.
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