Miscellaneous Properties of Relatively Free Groups

We classify the groups that are the inverse limits of sequences of free groups of finite rank. Except free groups of finite rank, there exist four groups that are the inverse limits of sequences of free groups of finite rank.

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

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

In this article we relate two different densities. Let $F_k$ be the free group of finite rank $k \ge 2$ and let $\alpha$ be the abelianization map from $F_k$ onto $ \mathbb{Z}^k$. We prove that if $S \subseteq \mathbb{Z}^k$ is invariant under the natural action of $SL(k, \mathbb{Z})$ then the asymptotic density of $S$ in $\mathbb Z^k$ and the annular density of its full preimage $\alpha^{-1}(S)$ in $F_k$ are equal. This implies, in particular, that for every integer $t\ge 1$, the annular density of the set of elements in $F_k$ that map to $t$-th powers of primitive elements in $\mathbb{Z}^k$ is equal to $\frac{1}{t^k\zeta(k)}$, where $\zeta$ is the Riemann zeta-function. An element $g$ of a group $G$ is called a \emph{test element} if every endomorphism of $G$ which fixes $g$ is an automorphism of $G$. As an application of the result above we prove that the annular density of the set of all test elements in the free group $F(a,b)$ of rank two is $1-\frac{6}{\pi^2}$. Equivalently, this shows that the union of all proper retracts in $F(a,b)$ has annular density $\frac{6}{\pi^2}$. Thus being a test element in $F(a,b)$ is an ``intermediate property'' in the sense that the probability of being a test element is strictly between $0$ and $1$.

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.

This paper is devoted to the computation of the space $$\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})$$ , where $$\Gamma $$ is a free group of finite rank $$n\ge 2$$ and $$H$$ is a subgroup of finite rank. More precisely we prove that $$H$$ has infinite index in $$\Gamma $$ if and only if $$\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})$$ is not trivial, and furthermore, if and only if there is an isometric embedding $$\oplus _\infty ^n\mathcal {D}({\mathbb {Z}})\hookrightarrow \mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})$$ , where $$\mathcal {D}({\mathbb {Z}})$$ is the space of bounded alternating functions on $${\mathbb {Z}}$$ equipped with the defect norm.

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

Characterizing a free group in its automorphism group

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 show that every right-angled Artin group AΓ defined by a graph Γ of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of Γ. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group AΓ has poly-free length 2 if and only if there exists an independent set of vertices D in Γ such that every cycle in Γ meets D at least twice. Finally, it is shown that AΓ is a semidirect product of two free groups of finite rank if and only if Γ is a finite tree or a finite complete bipartite graph. All of the proofs of the existence of polyfree structures are constructive.

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.

A Presentation for the Automorphism Group of a Free Group of Finite Rank

Let F 2 = 〈a, b〉 be the free group of rank 2 . We show that Aut F 2 has four conjugacy classes of elements of order 2 and one conjugacy class each of elements of order 3 and 4 . Implicit in the proof of these statements is an algorithm for deciding in which conjugacy class a periodic element belongs. From an analysis of GL(2, Z) ≅ Aut F 2/Inn F 2 , the orders of periodic elements are restricted to 2, 3, 4 and 6 , but Aut F 2 has no element of order 6 .

Let \(F_n\) be a free group of rank \(n\), with free generating set \(X\). A subset \(D\) of \(F_n\) is a Distinct Difference Configuration if the differences \(g^{-1}h\) are distinct, where \(g\) and \(h\) range over all (ordered) pairs of distinct elements of \(D\). The subset \(D\) has diameter at most \(d\) if these differences all have word length at most \(d\). When \(n\) is fixed and \(d\) is large, the paper shows that the largest distinct difference configuration in \(F_n\) of diameter at most \(d\) has size approximately \((2n-1)^{d/3}\).Mathematics Subject Classifications: 05B10, 20E05Keywords: Difference sets, distinct difference configurations, free groups, combinatorial designs