Unitary units and skew elements in group algebras

The fundamental group of an oriented surface of genus n with k boundary surfaces

Let F be the free group of rank 2, generated by x and y, and let w(x, y) ∈ F′ be a non-trivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y]n as a product of [n/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))2 cannot be written as one w-word and if g ≠ 1 ∈ w(F) then show that the minimal number of w-words required to express gn as their product tends to infinity with n. Other results for free groups of higher rank are also presented.

8046 INVITED Symptom occurrence, intensity and distress in patients during conditioning and early post-transplant period – implications for nursing

T. Bandman and Yu. G. Zarhin have proved that for every word $$w \in F_n, w\notin F_n^2$$, the corresponding word map $${\tilde{w}}: \mathrm {PGL}_2^n(K)\rightarrow \mathrm {PGL}_2(K)$$ is surjective if K is an algebraically closed field of characteristic zero (here $$F_n$$ is a free group of rank n and $$F_n^i$$ is ith member of the derived series). For words $$w\in F_n^2, n >1,$$ which are not decomposable into a product $$w = w_1w_2$$ of two words with independent variables, there are only two examples when the corresponding word map for the algebraic group $$\mathrm {PGL}_2$$ is surjective. In this paper, we construct infinite recursive sequences of indecomposable words $$\{w_m\}_{m\in {{\mathbb {N}}}}$$ in $$F_2$$ such that the word maps $${\tilde{w}}_m$$ are surjective for the algebraic group $$\mathrm {PGL}_2$$ and, if $$w_m\in F_2^i$$, then $$w_{m+1} \in F_2^{i+1}$$. Also, we construct infinite recursive sequences of indecomposable words $$\{w^\prime _m\}_{m\in {{\mathbb {N}}}}$$ in $$F_3$$ such that word maps $${\tilde{w}}^\prime _m$$ are surjective for the algebraic group $${{\text {SL}}}_2$$ and, if $$w^\prime _m\in F_3^i$$, then $$w^\prime _{m+1} \in F_3^{i+1}$$.

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras [Formula: see text] which are strongly Lie nilpotent of index 14.

Let F 2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for two given elements u , v of F 2 , u and v are translation equivalent in F 2 , that is, whether or not u and v have the property that the cyclic length of ( u ) equals the cyclic length of ( v ) for every automorphism of F 2 . This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edn. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) , Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.] for the case of F 2 .

Let kG be the group algebra of the non abelian locally finite group G over the non absolute field k of characteristic \(\ne 2\), and let \(^{*}\) be an involution of kG, extended linearly from an involution of G. We give necessary and sufficient conditions for kG to contain free pairs of \(^{*}\)-symmetric, and \(^{*}\)-unitary units.

We study the unitary subgroupV *(F2 G) in the group algebras F2 Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement to Gin V *(F2 G) if and only if Gis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describe V *(F2 G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V *(F2 G).

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal has already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

In this paper, we study the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for describing irreversible processes. The particular case of extensions parametrized by normal states is discussed in [1, 2]. The construction used below is similar to that of self-ajoint extensions for densely-defined, bounded below, symmetric operators in a Hilbert space. The dynamic semigroup, unlike the unitary operator group, acts not on elements of a Hilbert space 7/but on elements of a topological operator algebra in a Hilbert space B(H); moreover, this semigroup is completely positive (see [3]). For such semigroups, direct methods of describing domains of generating mappings are absent; this complicates the parametrization of extensions of the minimal generating mapping. It turns out that the range of the resolvent of the minimal dynamic semigroup belongs to the complement of the zero set (the kernel) of the limit mapping QOO(.) associated with the formal generating mapping of the dynamic semigroup. For this reason, we first consider general properties of the zero set for a completely positive mapping (see Sec. 1). In what follows, the C*-algebra generated by positive roots of a mapping Q(-) plays the central role. In Sec. 2, we prove that this algebra has the following property: it is maximal in the class of algebras that contain an element X if and only if X* X and X X* are elements of the algebra. In Sec. 3, the mentioned property is extended to the set of positive operators that are zeros of the limit mapping QOO(.) _- s-lira Qn (.). We prove that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal C*-Mgebra generated by positive roots of the mapping Q~(-) associated with the semigroup. In Sec. 4, we give a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Sec. 5, we prove the characteristic theorem for completely positive mappings with given ranges, which enables us to consider the set of conservative extensions as a *-weakly compact convex manifold in/3+(/2(7-~)), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements T in extreme points of E C T establishes a one-to-one correspondence between classes of equivalent normed measures on E and conservative extensions of the minimal dynamic semigroup; in this case, E plays the role of boundary and the class of equivalent measures plays the role of boundary condition for a conservative extension.

Lie properties of symmetric elements in group rings

Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in Fn whenever g, h ∈ Fn are translation equivalent in Fn, and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.].

For a finitely generated group G, we study deformation spaces of metric G-trees, which are analogues of the Teichmuller spaces of surfaces for group actions on trees. Deformation spaces of metric G-trees generalize Culler-Vogtmann’s Outer space, the deformation space of free actions on trees, which has proven to be immensely useful in the study of Out(Fn), the outer automorphism group of the free group of rank n ≥ 2. Let D be a deformation space of metric G-trees. The group of positive real numbers R>0 acts on D by scaling the metrics on the trees and we define the projectivized deformation space as the quotient PD = D/R>0. The outer auto- morphism group Out(G) contains a certain subgroup OutD(G) that acts on D and PD by precomposing the G-actions on the trees. In Chapter 1, we present a complete argument that under certain assumptions the projectivized deformation space PD is a model for the classifying space of OutD(G) for a family of subgroups. In Chapter 2, we introduce an asymmetric pseudometric on PD that generalizes the asymmetric Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmuller space. Making use of the Lipschitz metric on PD, we prove existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag-Solitar groups that contain no solvable Baumslag-Solitar group BS(1, n) with n ≥ 2. In Chapter 3, we define the higher holomorphs Aut(G, k), k ∈ N, which are “higher-pointed” variants of the automorphism group Aut(G). Following the construction of the spine of Outer space, we construct a family of simplicial complexes S(PD, k), k ∈ N on which certain subgroups AutD(G, k) ≤ Aut(G, k) act and we show that these complexes are always contractible.

We consider free groups of finite rank and write F k for the absolutely free group of rank k, freely generated by f = {f1, ..., f k }. An automorphism of F k maps a verbal subgroup U(F k ) onto itself; hence