Continuous control and the algebraic L-theory assembly map

The behavior of Nil-groups under localization and the relative assembly map

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

Let X be a mathbb {Q}-factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group mathrm{Pic}(X) in the group mathrm{Cl}(X) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of mathrm{Pic}(X) in mathrm{Cl}(X) is contained in a free part of the latter group.

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C ^* -algebras that are KK -equivalent to \mathbb C and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

The shuffle algebra is a free commutative algebra over the set of Lyndon words; this result is presented in Section 6.1, together with a precise identity on the shuffie product of Lyndon words, which implies that actually there is a canonical structure of algebra of divided powers. In Section 6.2 a remarkable presentation of the shuffle algebra is given; the generators are the nonempty words, and the relations the nontrivial shuffle products. In Section 6.3 we introduce subword functions on the free group, the Magnus transformation of the free group, the algebra structure on the module of subword functions, and the fact that this algebra is generated by the particular subword functions corresponding to Lyndon words. The main tools arc the concept of representative, or recognizable functions on the free group, and the infiltration product of Chen, Fox and Lyndon. Section 6.4 presents the commutator calculus of P. Hall, and its generalizations. There are many results involving the lower central series of the free group, the Magnus transformation, and the algebra of subword functions.

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in Nickel. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented by Nickel.

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

Let Γ < SL n ( $$\mathbb{Z}$$ ) be a subgroup of finite index, where n ≥ 5. Suppose Γ acts continuously on a manifold M, where π1(M) = $$\mathbb{Z}$$ n , preserving a measure that is positive on open sets. Further assume that the induced Γ action on H 1(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ : M → $$\mathbb{T}$$ n that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n &lt; 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I &lt; 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l &gt; 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l &gt; 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partial\Gamma=cd_L\Gamma-1 $$ connecting the cohomological dimension of a group $\Gamma$ with the cohomological dimension of its boundary $\partial\Gamma$. In [Be] Bestvina introduced a notion of $\sZ$-structure on a discrete group and noticed that his formula holds true for all torsion free groups with $\sZ$-structure. Bestvina's notion of $\sZ$-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivially is not valid for groups with torsion, we show that it still holds in the following modified form: {\it The cohomological dimension of a $\sZ$-boundary of a group $\Gamma$ equals its global cohomological dimension for every PID $L$ as the coefficient group} $$ \dim_L\partial\Gamma=gcd_L(\partial\Gamma). $$ Using this formula we show that the cohomological dimension of the boundary $\dim_{L}\partial\Gamma$ is a quasi-isometry invariant of a group.