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For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $$\mathbb {S}^n$$ , for $$n\ge 2$$ . Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in $$\mathbb {S}^n$$ , then it is the Julia set of a hyperbolic UQR map $$f:\mathbb {S}^N \rightarrow \mathbb {S}^N$$ where $$N=n$$ if $$n=2$$ and $$N=n+1$$ otherwise.

A locally compact contraction group is a pair ( G , α ) , where G is a locally compact group and α : G → G an automorphism such that α n ( x ) → e pointwise as n → ∞ . We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups ( G , α ) which are central extensions { 0 } → F p ( ( t ) ) → G → F p ( ( t ) ) → { 0 } of the additive group of the field of formal Laurent series over F p = Z / p Z by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.

A locally compact group G is said to be approximable by totally disconnected subgroups if there is a sequence of closed totally disconnected subgroups $$(\Gamma _n)_{n\in {\mathbb N}}$$ of G such that for any non-empty open set O of G, there exists an integer k such that $$O\cap \Gamma _n \ne \varnothing $$ , for every $$n\ge k$$ . In this paper, we prove that every pro-torus is approximable by profinite subgroups and we provide a characterization of the approximation of compact groups. Furthermore, we show that every compactly generated locally compact abelian group is approximable by totally disconnected subgroups. As an application, a study of the Chabauty space of closed totally disconnected subgroups is given.

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; they extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N . We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

For a locally compact metrizable group G, we consider the action of $$\mathrm{Aut}(G)$$ on $$\mathrm{Sub}_G$$ , the space of all closed subgroups of G endowed with the Chabauty topology. We study the structure of groups G admitting automorphisms T which act expansively on $$\mathrm{Sub}_G$$ . We show that such a group G is necessarily totally disconnected, T is expansive and that the contraction groups of T and $$T^{-1}$$ are closed and their product is open in G; moreover, if G is compact, then G is finite. We also obtain the structure of the contraction group of such T. For the class of groups G which are finite direct products of $${\mathbb {Q}}_p$$ for distinct primes p, we show that $$T\in \mathrm{Aut}(G)$$ acts expansively on $$\mathrm{Sub}_G$$ if and only if T is expansive. However, any higher dimensional p-adic vector space $${\mathbb {Q}}_p^n$$ , ( $$n\ge 2$$ ), does not admit any automorphism which acts expansively on $$\mathrm{Sub}_G$$ .

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as – decomposing them into local products of their Sylow p- subgroups, – providing new descriptions of periodic abelian torsion groups, and of – periodic abelian divisible groups and their torsion-free and their torsion components, – reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent p-groups, – providing a definition of a general p-rank for all locally compact abelian p-groups.

Let ϕ : G → H {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair ( G , ϕ - 1 ⁢ ( L ) ) {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

Fractal theory is the study of irregularity which occurs in natural objects. It also enables us to see patterns in the highly complex and unpredictable structures resulting from many natural phenomena, using self-similarity property. The most common mathematical method to generate self-similar fractals is using an iterated function system (IFS). This paper discusses separation properties of finite products of hyperbolic IFSs. Characterizations for totally disconnected and overlapping product IFSs are obtained. A method to generate an open set which satisfies the open set condition for a totally disconnected IFS is given. Some necessary and sufficient conditions for a product IFS to be just touching are discussed. Also, Type 1 homogenous IFSs are introduced and its separation properties in terms of the separation properties of coordinate projections are explained towards the end.

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we construct a hyperbolic graph on $(A, {\mathcal D})$ and show that $K$ can be identified with the hyperbolic boundary. Moreover, if $(A, {\mathcal D})$ safisfies the open set condition, we also prove that two totally disconnected integral self-affine sets are Lipschitz equivalent if an only if they have the same $w$-Hausdorff dimension, that is, their digit sets have equal cardinality. We extends some well-known results in the self-similar sets to the self-affine sets.

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given., Comment: 17 pages

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group $\Gamma$ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center $\Gamma_{\rm fc}$ of $\Gamma$ (that is, the normal sugbroup of $\Gamma$ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of $\Gamma_{\rm fc}$. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II., Comment: 26 pages

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio&rsquo, s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

In this paper we investigate the block numerical range and the existences of estimable decompositions of bounded linear operators on a separable Hilbert space. By using spectral measure, we show that there exists an estimable decomposition for the spectrum of every bounded normal operator. Furthermore, the corresponding result also holds for hyponormal operators with totally disconnected spectra. Finally, we obtain that for spectral operator, there exists an estimable decomposition, under quasi-nilpotent equivalence.

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C &lowast, algebras of the groupoids of germs have a unique tracial state.

We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of Bader, Furman and Sauer. We apply this criterion to show that the categories constructed from totally disconnected groups by Arano and Vaes have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda, Comment: 44 pages, 1 figure. v2: Minor corrections. Added a remark and an example

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1., Comment: Abstract slightly reformulated

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings R, in particular for R = Z and R = Q . We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both FP n -properties and F n -properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.

International audience; We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.

In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq ��}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling" where an attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the same period. We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of "holomorphic motions" to our context. In addition, we prove that there is an "infinitely renormalizable" tangent map $T_{t_\infty}$. It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under $T^2_{t_\infty}$. The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect and unbounded., 52 pages, 15 figures

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B)., more detailed version - some corrections have been made -title has been changed

A compact, Hausdorff and totally disconnected loop is called a boolean loop. In this note, we prove that a boolean loop with compact left inner mapping group is profinite.

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A.Willis, Math. Ann. 361 (2015), 403–442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.

There is no way to build a nontrivial Markov process having continuous trajectories on a totally disconnected fractal embedded in the Euclidean space. Accordingly, in order to delineate the diffusion process on the totally disconnected fractal, one needs to relax the continuum requirement. Consequently, a diffusion process depends on how the continuum requirement is handled. This explains the emergence of different types of anomalous diffusion on the same totally disconnected set. In this regard, we argue that the number of effective spatial degrees of freedom of a random walker on the totally disconnected Cantor set is equal to n s p = [ D ] + 1 , where [ D ] is the integer part of the Hausdorff dimension of the Cantor set. Conversely, the number of effective dynamical degrees of freedom ( d s ) depends on the definition of a Markov process on the totally disconnected Cantor set embedded in the Euclidean space E n ( n ≥ n s p ) . This allows us to deduce the equation of diffusion by employing the local differential operators on the F α -support. The exact solutions of this equation are obtained on the middle-ϵ Cantor sets for different kinds of the Markovian random processes. The relation of our findings to physical phenomena observed in complex systems is highlighted.

The author has previously associated to each commutative ring with unit $R$ and \'etale groupoid $\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okni\'nski of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras., Comment: Fixed a misstatement in the proof of Proposition 8 and corrected some minor typos/wording

In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group. When the building is locally finite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact open subgroups of the universal groups as a limit of generalized wreath products., 49 pages, 2 figures

We study the interplay between Steinberg algebras and skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorff ample groupoid, can be seen as a partial skew inverse semigroup ring.

The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute some computations and complements to the general theory of $L^2$-Betti numbers of totally disconnected groups.

The study of Lipschitz equivalence of fractals is a very active topic in recent years, but there are very few results on fractals which is not totally disconnected. In this paper, using finite state automata and the angle separation property, we prove that for a class of self-similar sets with two-state neighbor automaton, two elements are Lipschitz equivalent if and only if they have the same Hausdorff dimension.

The iteration of the map $$z \mapsto \lambda + \tan z$$ on the complex plane is investigated for each complex number $$\lambda $$ . It is shown that its Fatou set always contains a completely invariant component and the Julia set is either totally disconnected or connected. A necessary and sufficient condition for totally disconnected Julia set is proved. When the Julia set is connected, it is seen that the Fatou set can consist of additional components, which can be completely invariant attracting domain, a Siegel disk, a two periodic attracting domain or parabolic domain. Examples of each case are given. Further, a necessary and sufficient condition is provided for the Julia set to be a Jordan curve passing through infinity. It is proved that the set of parameters corresponding to Jordan curve Julia sets is connected, whereas the set of those corresponding to totally disconnected Julia sets is disconnected.

We introduce the notion that a zero-dimensional compact space $L$ is a \emph{Boolean image} of an arbitrary compact space $K$. When $K$ is also zero-dimensional, this just means that $L$ is a continuous image of $K$. However, a number of interesting questions arise when we consider connected compacta $K$., Revised version

For each finite set S of prime numbers there exists a unique completion ℚ S of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2(ℚ S ). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups.

Fractal sets typically have very complex geometric structures, and a fundamental problem in fractal geometry is to characterize how “similar” different fractal sets are. The Lipschitz equivalence of fractal sets is often used to classify fractal sets that are geometrically similar. Interesting links between Lipschitz equivalence and algebraic properties of contraction ratios for self-similar sets have been uncovered and widely analyzed. However, with the exception of very few papers, the study of Lipschitz equivalence has largely focused on totally disconnected self-similar sets. For connected self-similar sets this problem becomes rather challenging, even for well known fractal models such as fractal squares. In this paper, we introduce geometric and topological methods to study the Lipschitz equivalence of connected fractal squares. In particular we completely characterize the Lipschitz equivalence of fractal squares of order 3 in which one or two squares are removed. We also discuss the Lipschitz equivalence of fractal squares of more general orders. Our paper is the first study of Lipschitz equivalence for nontrivial connected self-similar sets, and it raises also some interesting questions for the more general setting.

Given a partial action of a discrete group G on a Hausdorff, locally compact, totally disconnected topological space X, we consider the correponding partial action of G on the algebra Lc(X) consisting of all locally constant, compactly supported functions on X, taking values in a given field K. We then study the ideal structure of the algebraic partial crossed product Lc(X)⋊G. After developing a theory of induced ideals, we show that every ideal in Lc(X)⋊G may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic version of the Effros–Hahn conjecture.

A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of \'Alvarez L\'opez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties., Comment: Minor corrections; the term `tame' changed to `stable'

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ and Radon probability measures on the vertex space $E^{0}$ which are, in a suitable sense, invariant under the action of the edge space $E^{1}$. It is shown that if $E$ has no cycles, then every tracial state on $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.

Let X be a set in ℝn with positive Lebesgue measure. It is well known that the spectrum of the algebra L∞(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒb(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τdis),where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒb(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L∞(X))), of the ideal of all functions in ℒb(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒb(X, ℂ).

We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer n such that every neighborhood of the identity contains n elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally 2-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally 1-generated but totally disconnected, strengthening Stevens' negative answer to Problem 160 from the Scottish book.

It is shown that, given a lattice H H in a totally disconnected, locally compact group G G , the contraction subgroups in G G and the values of the scale function on G G are determined by their restrictions to H H . Group theoretic properties intrinsic to the lattice, such as being periodic or infinitely divisible, are then seen to imply corresponding properties of G G .

We construct dendrites with endpoint sets isometric to any totally disconnected compact metric space. This allows us to embed zero-dimensional dynamical systems into dendrites and solve a problem regarding Li-Yorke and distributional chaos., 14 pages

In this paper, we establish a topological version of the notion of an Eilenberg&ndash, Mac Lane space. If X is a pointed topological space, &pi, 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 &rarr, X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication &pi, 1 ( X ) &times, &pi, 1 ( X ) &rarr, 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 &rarr, X and the product &pi, 1 ( X ) with compactly generated topologies to see that &pi, 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg&ndash, Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, &pi, 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer&rsquo, s. We also show that analogues of the Mayer&ndash, Vietoris sequence and Seifert&ndash, van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world.

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is non-compact and every closed normal subgroup of $G_i$ is discrete or cocompact (e.g. $G_i$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\Gamma$ with dense projections is finite. The same result holds if $\Gamma$ is non-uniform, provided $G$ has Kazhdan's property (T). We show that for any compact subset $K \subset G$, the collection of discrete subgroups $\Gamma \leq G$ with $G = \Gamma K$ and dense projections is uniformly discrete, hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $Aut(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups., Comment: v1: 51 pages. v2: 52 pages. v3: final version

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

In this paper, we prove that the algebra of an \'etale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal \'etale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an \'etale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of \'etale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and \'etale groupoids with simple algebras outside of a prescribed set of prime characteristics., Comment: Revisions after a referee report. We define the essential algebra of an ample groupoid as the quotient of the Steinberg algebra by its ideal of singular functions and we prove that the essential algebra is simple if and only if the groupoid is minimal and topologically free. When the singular ideal vanishes, one recovers the simplicity result of the previous version of the paper

Our main objective is to study g-reversible groups introduced in Chatyrko and Shakhmatov (2020) [3] . We first give a necessary and sufficient condition for a locally compact group with an open compact subgroup to be g-reversible, it can be applied to give complete solutions of Questions 8.5, 9.7, 9.8, and Problem 9.6 in the previous article [3] . We also investigate g-reversibility of some precompact abelian groups, in particular, we give a pseudocompact g-reversible abelian group which is not minimal, which answers Question 11.5 in the same article [3] negatively. We give descriptions of connected (or totally disconnected) compact abelian groups of which all subgroups are g-reversible. Finally, we give a characterization of the abelian groups which admit hereditarily g-reversible compact group topologies.

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property ( T ) (T) . We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}.

We study automorphisms $\alpha$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $\alpha^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $\alpha$ on $G/N$ for an $\alpha$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_\alpha$ consisting of all $x$ in $G$ such that $\alpha^n(x) \to e$ as $n$ tends to infinity. If $\alpha$ is expansive, then $W := U_\alpha U_{\alpha^{-1}}$ is an open identity neighbourhood in $G$. We give examples where $W$ fails to be a subgroup. However, $W$ is a nilpotent open subgroup whenever $G$ is a closed subgroup of a general linear group over the $p$-adic numbers. Further results are devoted to the divisible and torsion parts of $U_\alpha$, and to the so-called "nub" $U_0$ of an expansive automorphism $\alpha$ (the intersection of the closures of $U_\alpha$ and $U_{\alpha^{-1}}$)., Comment: LaTeX, 32 pages; v3: minor improvements