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We prove that for every r > 0 r>0 if a nonpositively curved ( p , q ) (p,q) -map M M contains no flat submaps of radius r r , then the area of M M does not exceed C r n Crn for some constant C C . This strengthens a theorem of Ivanov and Schupp. We show that an infinite ( p , q ) (p,q) -map which tessellates the plane is quasi-isometric to the Euclidean plane if and only if the map contains only finitely many nonflat vertices and faces. We also generalize Ivanov and Schupp’s result to a much larger class of maps, namely to maps with angle functions.

The authors correct an error in Proposition 3.24 of their work Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505.

Recently Vaughan Jones showed that the R. Thompson group $F$ encodes in a natural way all knots, and a certain subgroup $\vec F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\vec F$. In particular we prove that the subgroup $\vec F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i, i=0,1,2,...$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\vec F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\vec F$ is irreducible., First draft, 14 pages; v2: Second draft, added Section 5 generalizing the main results from $n=2$ to arbitrary $n>1$. v3: Added a subsection about natural homomorphisms and caret replacement homomorphisms v4: Added a section proving, in particular, that the Thompson index of a link is bounded by a linear function in the number of crossings in a link diagram, added some open problems

The isoperimeric spectrum consists of all real positive numbers $\alpha$ such that $O(n^\alpha)$ is the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf{P=NP}$ and even a formally weaker conjecture., Comment: 3 pages

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

We show how a recent result of Hrushovsky implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent., 3 pages

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic cone of a geodesic metric space is homogeneous and has cut points, then it is the universal tree-graded space with pieces - maximal connected subsets without their own cut points. Thus it is completely determined by its collection of pieces., 24 pages; v2: a reference added, we also made some changes suggested to us by Yves de Cornulier; v3: the paper is accepted in IJAC

We prove that with probability tending to 1, a 1-relator group with at least 3 generators and relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions, bridges and excursions in R^k., 11 pages. Comments are welcome. v2: Added a new probabilistic model: counting 1-relataed groups up o isomorphism, not words as before

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.

The aim of this note is to answer a question by Guoliang Yu of whether the group $EL_3(Z)$, where $Z$ is the free (non-commutative) ring, has any faithful linear representations over a field. We prove, in particular, that for every (unitary associative) ring $R$, the group $EL_3(R)$ has a faithful finite dimensional complex representation if and only if $R$ has a finite index ideal that has a faithful finite dimensional complex representation., Comment: 8 pages, no figures; v3: a question about EL_2 from the first version is answered

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ? 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ? m + 1 (resp., k ? m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ? 3.

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir)., 44 pages

We give an example of a finitely presented group G with two non-π1-equivalent asymptotic cones.

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$., 20 pages; v2: fixed some misprints, filled a gap in the proof of Theorem 4.1, added Remark 4.1 that Homeo^+(R) and many subgrioups of that group are quasi-residually finite; v3: Section 5 added, final version accepted to Transactions of the AMS

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n2log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n2log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem.

We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group $F$ is equal to 1/2, the Hilbert space compression of the restricted wreath product $Z\wr Z$ is between 1/2 and 3/4, and the Hilbert space compression of $Z\wr (Z\wr Z)$ is between 0 and 1/2. In general, we find a relationship between the growth of $H$ and the Hilbert space compression of $Z\wr H$., Comment: 20 pages, Theorem 1.13 and Lemma 3.7. are new

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

For every finitely generated recursively presented group G {\mathcal G} we construct a finitely presented group H {\mathcal H} containing G {\mathcal G} such that G {\mathcal G} is (Frattini) embedded into H {\mathcal H} and the group H {\mathcal H} has solvable conjugacy problem if and only if G {\mathcal G} has solvable conjugacy problem. Moreover, G {\mathcal G} and H {\mathcal H} have the same r.e. Turing degrees of the conjugacy problem. This solves a problem by D. Collins.

We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 n\gg 1 by a cyclic group, so it satisfies the identity [ x , y ] n = 1 [x,y]^n=1 .

We produce two examples demonstrating that an ascending HNN extension of a finitely generated residually finite group need not be residually finite. The first example has very few quotients. The second example is not Hopfian.

This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity" from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a simple "non-amenability-like" property that follows from RD, and give an easy example of a group without RD and without any amenable subgroup with superpolynomial growth, some misprints in other sections are corrected., 27 pages; v2: Section 2 corrected, added a reference to Olshansii's preprint arXiv:1406.0336 in Section 4. v3: Section 3.6 on graph products of groups is added. v3: Small correction in the proof of Theorem 2.3, estimate slightly improved

The Tarski number of a non-amenable group G is the minimal number of pieces in a paradoxical decomposition of G. In this paper we investigate how Tarski numbers may change under various group-theoretic operations. Using these estimates and known properties of Golod-Shafarevich groups, we show that the Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We also use the cost of group actions to show that there exist groups with Tarski numbers 5 and 6. These provide the first examples of non-amenable groups without free subgroups whose Tarski number has been computed precisely., 26 pages. v4: We have added the result about the existence of groups with Tarski number 5 which previously appeared in a separate paper of the second author arXiv:1406.2097. Section 5 has been rewritten. Minor changes in Appendix B. We also replaced left group actions by right group actions

We start with an old problem from number theory whose solution leads to the first example of an infinite cube-free word, thus showing that the word x 3 is avoidable. Then we shall describe all avoidable words.

This chapter introduces the main characters that will appear in this book: sets, words, graphs, automata, rewriting systems, various kinds of (universal) algebras, varieties, free algebras (including free semigroups and groups) and subshifts. We also introduce the main properties of algebras that we are interested in: the Burnside property, the finite basis property, properties of the growth function and the growth series, etc.

We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups. Résumé: Nous établissons un lien entre le problème du calcul de l'adhéerence d'un sous-groupe finiment engendré du groupe libre dans la topologie pro-V, oú V est une pseudovariété de groupes finis, et un probléme d'extension pour les automates inversifs qui peut être énoncé de la faç con suivante: étant données des transformations partielles injectives d'un ensemble fini, peuvent-elles être étendues en des permutations qui engendrent un groupe dans V? Les deux problèmes sont équivalents si V est fermée par extensions. Nous intéressant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adhérence pro-p d'un sous-groupe finiment engendré du groupe libre en temps polynomial et pour calculer effectivement sa clôture pro-nilpotente. Enfin nous appliquons nos résultats à un problème de théorie des monoïdes finis, celui de de l'appartenance dans les pseudovariétés de monoïdes inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariété de groupes.

It is shown that every ascending HNN extension of a nitely generated free group is Hopan. An important ingredient in the proof is that under certain hypotheses on the group H ,i fG is an ascending HNN extension of H, then cd(G )= cd(H )+1 . 1. Statement of results Ag roupG is Hopan, provided that every surjective endomorphism of G is an automorphism. This notion originated in connection with Hopf’s question of whether a degree 1 map from a closed manifold to itself must be a homotopy equivalence. While it is easy to give examples of innitely generated groups which are not Hopan, for some time it was an open question as to whether every nitely generated group is indeed Hopan. The rst nitely generated example was given by B. Neumann in [13], and shortly thereafter the following nitely presented example was given by G. Higman in [8]: ha;s;tj a t = a 2 ;a s = a 2 i

We survey recent results about asymptotic functions of groups, obtained by the authors in collaboration with J.-C. Birget, V. Guba and E. Rips. We also discuss methods used in the proofs of these results.

We prove that there exists an amalgam of two finite 4-nilpotent semigroups such that the corresponding amalgamated product has an undecidable word problem. We also show that the problem of embeddability of finite semigroup amalgams in any semigroups and the problem of embeddability of finite semigroup amalgams into finite semigroups are undecidable. We use several versions of Minsky algorithms and Slobodskoj's result about undecidability of the universal theory of finite groups.

In this article we show that the Dehn function of a nontrivial free product of groups is equivalent to its subnegative closure.

A subset M of a universal algebra A is called finitely separated in A if, for any element x∈A/M, there exists an homomorphism ϕ of A into a finite algebra, for which ϕ(x)∋ϕ(M). A ring is said to be S-separable (R-separable) if its subrings (resp., right ideals) are all finitely separated in it. We give equational (in the language of identities) and indicator (in the language of “prohibited” rings) characterizations of varieties consisting of S-separable (R-separable) rings. Moreover, varieties are described in which, not all, but finitely generated rings only share the properties mentioned.

We prove that there is no algorithm to decide, given a finite semigroup S and two elements a, b∈S, whether there exists a bigger finite semigroup T>S where a divides b and b divides a. This solves a thirty years old problem by John Rhodes.

We construct a finitely based variety of algebras with two binary operations where the set of subalgebras of finite simple algebras is not recursive.

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic finii e semigroups are irreducible for join, for semidirect product and for Mal’cev product. In particular, these pseudovarieties do not admit maximal proper subpseudovarieties. More generally, analogous results are proved for the pseudovariety of all finite semigroups all of whose subgroups are in a fixed pseudovariety of groups H, provided th.it H is closed under semidirect product.

An algebra is inherently non-finitely (Q-)based if it is not a member of any locally finite (quasi-)variety, whose (quasi-)identities are finitely based. We prove that no finite semigroup is inherently non-finitely Q-based. This is in marked contrast to the case of varieties, where there are many inherently non-finitely based finite semigroups which have all been described by the second author.

A semilatticeS isrepresentable by subspaces ofRk if, to eachx eS we can assign a subspace\(V_x \subseteq R^k\) so thatx Λy=z inS if and only if\(V_x \cap V_y = V_z\). Every height-2 semilattice is representable inR2. We show that for everyk there is a height-3 semilattice which is not representable by subspaces ofRk.

We construct a finitely presented group with undecidable word problem and with Dehn function bounded by a quadratic function on an infinite set of positive integers., 108 pages, 24 figures

We establish a one-to-one “group-like” correspondence between congruences on a free monoid X* and so-called positively self-conjugate inverse submonoids of the polycyclic monoid P(X). This enables us to translate many concepts in semigroup theory into the language of inverse semigroups.

Inverse semigroups are considered as algebras with two operations: multiplication and inversion. We prove that there are no inherently non-finitely based finite inverse semigroups. In constrast, if we consider inverse semigroups as semigroups (one binary operation) then it was proved by the author that almost all finite inverse semigroups are inherently non-finitely based.