A group $G$ given by a presentation $G = < \mathcal A \| \mathcal R >$ is called weakly finitely presented if every finitely generated subgroup of $G$, generated by (images of) some words in $\mathcal A^{\pm 1}$, is naturally isomorphic to the subgroup of a group $G_0 = < \mathcal A_0 \| \mathcal R_0>$, where $\mathcal A_0 \subseteq \mathcal A$, $\mathcal R_0 \subseteq \mathcal R$ are finite, generated by (images of) the same words. In the article, weakly finitely presented periodic groups which are not locally finite are constructed.
We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf $G$-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on $\ell_{\infty}$-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.
We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ and rank $m >1$ in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.
We generalize a well known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization will be used later to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.
Rodaro and Silva proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences bet…
In this paper, a group is called weakly amenable if its left regular representation is not uniformly isolated from the trivial representation. First examples of finitely generated non-amenable weakly amenable groups are constructed.
A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck--Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of $S$ to be finitely generated.
We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.
Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups tha…
We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the quotient of $G$ by the normal subgroup generated by $g^n$ is large for all but finitely many $n\in \mathbb Z$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $(p_1, p_2, ...)$, there exists an infinite finitely generated periodic group $Q$ with descending normal series $Q=Q_0\rhd Q_1\rhd ... $, such t…
We prove that finitely generated virtually free groups are stable in permutations. As an application, we show that almost-periodic almost-automorphisms of labelled graphs are close to periodic automorphisms.
A group $Γ$ is said to be periodic if for any $g$ in $Γ$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose …
For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…
We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group. We show that if the associated Schreier graphs are linearly repetitive, then the group is of intermediate growth. In particular, this gives first examples of simple groups of intermediate growth.
Let $M\subset B(\mathcal H)$ be a von Neumann algebra acting on the Hilbert space $\mathcal H$. We prove that $M$ is finite if and only if, for every $x\in M$ and for all vectors $ξ,η\in\mathcal H$, the coefficient function $u\mapsto \langle uxu^*ξ|η\rangle$ is weakly almost periodic on the topological group $U_M$ of unitaries in $M$ (equipped with the weak or strong operator topology). The main device is the unique invariant mean on the $C^*$-algebra $\operatorname{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^*ξ|…
We prove that a saturated weakly branch group $G$ has the property $R_\infty$ (any automorphism $ϕ:G\to G$ has infinite Reidemeister number) in each of the following cases: 1) any element of $Out(G)$ has finite order; 2) for any $ϕ$ the number of orbits on levels of the tree automorphism $t$ inducing $ϕ$ is uniformly bounded and $G$ is weakly stabilizer transitive; 3) $G$ is finitely generated, prime-branching, and weakly stabilizer transitive with some non-abelian stabilizers (with no restrictions on automorphisms). Some related facts and generalizations are proved.
We completely characterise when the sequence of free subgroup numbers of a finitely generated virtually free group is ultimately periodic modulo a given prime power.
For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-c…
We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a $\log$-uniformly continuous normal order is isomorphic to $(\mathbb{Z}, +)$.
We show that for any countable group, any free probability measure preserving action of the group weakly contains all Bernoulli actions of the group. It follows that for a finitely generated groups, the cost is maximal on Bernoulli actions and that all free factors of i.i.d.-s the group have the same cost. We also show that if a probability measure preserving action f is ergodic, but not strongly ergodic, then f is weakly equivalent to f\timesI where I denotes the trivial action on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy.
We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. García-Ramos and H. Li, and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is pre-injective. This together with an earlier result of…
A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Brick's qsf property, which amounts to finding an exhaustion approximable by finite simply connected complexes, and also to the tame combability introduced and studied by Mihalik and Tschantz. We further observe that a number of standard constructions in group theory yield qsf groups and analyze sp…
We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.
Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--Sàndor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this arti…
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\langle G(X),ρ_e\cupρ_s\rangle$ and its structure is studied. Although each of the sets $G(X)$, $ρ_e$, and $ρ_s$ is infinite for $|X|\geq 2$, we show that the word problem …
It is well known that if $G$ admits a f.g. subgroup $H$ with a weaklyaperiodic SFT (resp. an undecidable domino problem), then $G$itself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove that we can replace the property "$H$ is a subgroup of $G$"by "$H$ acts translation-like on $G$", provided $H$ is finitely presented.In particular:* If $G\_1$ and $G\_2$ are f.g. infinite groups, then $G\_1 \times G\_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-g…
