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Ateş, Fırat (Balikesir Author), For arbitrary monoidsAandB, in Cevik et al. (Hacet J Math Stat 2019:1-11, 2019), it has been recently defined an extended version of the general product under the name ofa higher version of Zappa products for monoids(orgeneralized general product)A(circle plus B) (delta)(sic)(psi) B-circle plus A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green's relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice., Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah G:1709-247-1440

A dessin d’enfant or dessin is a bicolored graph embedded into a Riemann surface. Acyclic dessins can be described analytically by preimages of Shabat polynomials and algebraically by their monodromy groups. We determine the Shabat polynomials and monodromy groups of planar acyclic dessins that are uniquely determined by their ramification types.

Let $\Gamma$ be a finite group. Consider the wreath product $G_n := \Gamma^n \rtimes S_n$ and the subgroup $K_n := \Delta_n \times S_n\subseteq G_n$, where $S_n$ is the symmetric group and $\Delta_n$ is the diagonal subgroup of $\Gamma^n$. For certain values of $n$ (which depend on the group $\Gamma$), the pair $(G_n, K_n)$ is a Gelfand pair. It is not known for all finite groups which values of $n$ result in Gelfand pairs. Building off the work of Benson--Ratcliff, we obtain a result which simplifies the computation of multiplicities of irreducible representations in certain tensor product representations, then apply this result to show that for $\Gamma = S_k, \ k \geq 5$, $(G_n,K_n)$ is a Gelfand pair exactly when $n = 1,2$., Comment: 6 pages; preliminary version, comments welcome

In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups., 27 pages

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds., Comment: 11 pages; final version to appear in Algebraic & Geometric Topology

We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath product constructed from it is unitarizable if and only if the acting group is amenable., Several improvements in the exposition. No new results added. To be published in the Bulletin of the Belgian Mathematical Society-Simon Stevin

International audience; In an earlier work, the author observed that Boolean inverse semigroups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are monoids of generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups.

In this paper we show that the mapping class groups of any surface with nonempty boundary form a simplicial group as the number of marked points varies. This extends the simplicial structure on braid groups of surfaces found by Berrick, Cohen, Wong and Wu. We use the simplicial maps to construct compatible normal forms for elements of the braid groups and mapping class groups of surfaces with boundary.

A detailed study of the characters of $\mathfrak{S}_{\infty}(T)$ , the wreath product of compact group $T$ with the infinite symmetric group $\mathfrak{S}_{\infty}$ , is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product $\mathfrak{S}_{n}(T)$ as $n\to\infty$ and its connection with characters of $\mathfrak{S}_{\infty}(T)$ . This paper takes a dual approach to these problems. We study harmonic functions on $\mathbb{Y}(\widehat{T})$ , the branching graph of the inductive system of $\mathfrak{S}_{n}(T)$ ’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of $\mathfrak{S}_{\infty}(T)$ , which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on $\mathbb{Y}(\widehat{T})$ and Martin integral expressions for harmonic functions.

We introduce the notion of the rank gradient function of a descending chain of subgroups of finite index and show that the lamplighter group [math] has uncountably many 2-chains (that is, chains in which each subsequent group has index 2 in the previous group) with pairwise different rank gradient functions. In doing so, we obtain some information on subgroups of finite index in the lamplighter group.

Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups., 4 pages

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete description of semidirect decompositions and values of minimal permutation degrees.

We study the asymptotics of the reducible representations of the wreath products G\wr S_q=G^q \rtimes S_q for large q, where G is a fixed finite group and S_q is the symmetric group in q elements; in particular for G=Z/2Z we recover the hyperoctahedral groups. We decompose such a reducible representation of G\wr S_q as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products., 19 pages

In this paper we investigate Uludag's method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups. In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamental groups. In the second part, we discuss some properties of groups which are preserved by these methods. Afterwards, we describe precisely the families of curves which can be obtained by applying the generalized methods to several types of plane curves. We also give an application of the general methods for constructing new Zariski pairs., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-21.abs.html

Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting a well-known argument for Z^d, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential., The presentation has been improved: some geometric group theory background has been added to make it more friendly to probabilists. The proofs are now self-contained (instead of relying on some other work of the second author)

Let $G, H$ be finite groups. We asymptotically compute $|\mathrm{Hom}(G, H\wr A_n)|$, thereby establishing a conjecture of T. Müller.

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, raised by Cohen [Whitehead torsion, group extensions, and Zeeman's conjecture in high dimensions, Topology, 16 (1977) 79--88]., Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper5.abs.html

A study is made of the action of the fundamental group of a bouquet of a circle and a 1-connected space on the higher homotopy groups. If the 1-connected space is a suspension space, it is shown, with the aid of a theorem of Hartley on wreath products of groups and the Hilton-Milnor theorem, that the action is residually nilpotent. An unsuccessful approach in the case of a general 1-connected space is discussed, as it has some interesting features.

We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves Alexander's bounds. Our approach, which does not use the subadditive theorem, is based on proving that the average distance is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its "level of geodesicity"., Comment: The proof of the main theorem contained a few mistakes that have been corrected. The presentation has also been improved

A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus., knot, link, operator group, virtual link, virtual genus

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x), Comment: 8 pages. v3: final version, with new presentation; to appear in the Bulletin of the BMS

Let [math] be a group of order [math] where [math] is prime and [math] . We give a strategy to enumerate the regular subgroups of [math] normalized by the left representation [math] of [math] . These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions [math] with [math] . We prove that every such regular subgroup is contained in the normalizer in [math] of the [math] -Sylow subgroup of [math] . This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order [math] , where [math] is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group.

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever $H$ also shares those conditions. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.

In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs., Comment: 19 pages