113 research outputs found
On the quasi-component of pseudocompact abelian groups
In this paper, we describe the relationship between the quasi-component q(G)
of a (perfectly) minimal pseudocompact abelian group G and the quasi-component
q(\widetilde G) of its completion. Specifically, we characterize the pairs
(C,A) of compact connected abelian groups C and subgroups A such that A \cong
q(G) and C \cong q(\widetilde G). As a consequence, we show that for every
positive integer n or n=\omega, there exist plenty of abelian pseudocompact
perfectly minimal n-dimensional groups G such that the quasi-component of G is
not dense in the quasi-component of the completion of G.Comment: minor revisio
Compact-like abelian groups without non-trivial quasi-convex null sequences
In this paper, we study precompact abelian groups G that contain no sequence
{x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G,
and x_n --> 0. We characterize groups with this property in the following
classes of groups:
(a) bounded precompact abelian groups;
(b) minimal abelian groups;
(c) totally minimal abelian groups;
(d) \omega-bounded abelian groups.
We also provide examples of minimal abelian groups with this property, and
show that there exists a minimal pseudocompact abelian group with the same
property; furthermore, under Martin's Axiom, the group may be chosen to be
countably compact minimal abelian.Comment: Final versio
Quasi-convex sequences in the circle and the 3-adic integers
In this paper, we present families of quasi-convex sequences converging to
zero in the circle group T, and the group J_3 of 3-adic integers. These
sequences are determined by an increasing sequences of integers. For an
increasing sequence \underline{a}=\{a_n\} of integers, put g_n=a_{n+1}-a_n. We
prove that:
(a) the set \{0\}\cup\{\pm 3^{-(a_n+1)} : n\in N\} is quasi-convex in T if
and only if a_0>0 and g_n>1 for every n\in N;
(b) the set \{0\}\cup\{\pm 3^{a_n} : n\in N\} is quasi-convex in the group
J_3 of 3-adic integers if and only if g_n>1 for every n\in N.
Moreover, we solve an open problem of Dikranjan and de Leo by providing a
complete characterization of the sequences \underline{a} such that
\{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in T. Using this result,
we also obtain a characterization of the sequences \underline{a} such that the
set \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in R.Comment: 19 page
Algebraic entropy in locally linearly compact vector spaces
We introduce algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as a natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in Giordano Bruno and Salce (Arab J Math 1:69\u201387, 2012). We show that the main properties continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem
Reflection principle characterizing groups in which unconditionally closed sets are algebraic
We give a necessary and sufficient condition, in terms of a certain
reflection principle, for every unconditionally closed subset of a group G to
be algebraic. As a corollary, we prove that this is always the case when G is a
direct product of an Abelian group with a direct product (sometimes also called
a direct sum) of a family of countable groups. This is the widest class of
groups known to date where the answer to the 63 years old problem of Markov
turns out to be positive. We also prove that whether every unconditionally
closed subset of G is algebraic or not is completely determined by countable
subgroups of G.Comment: 14 page
Endomorphisms of Abelian Groups with Small Algebraic Entropy
We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means ). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group , where an automorphism ϕ with must have all eigenvalues in the open circle of radius 2, centered at 0 and ϕ must leave invariant a lattice in , i.e., be essentially an automorphism of . In particular, all eigenvalues of an automorphism ϕ with must be roots of unity. This is a particular case of a more general fact known as Algebraic Yuzvinskii Theorem. We discuss other particular cases of this fact and we give some applications of our main results
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