We study homomorphisms from Out(F_3) to Out(F_5), and GL(m,K) for m < 7,
where K is a field of characteristic other than 2 or 3. We conclude that all
K-linear representations of dimension at most 6 of Out(F_3) factor through
GL(3,Z), and that all homomorphisms from Out(F_3) to Out(F_5) have finite
image.

We derive a recurrence relation for the number of simple vertex-labelled
bipartite graphs with given degrees of the vertices and use this result to
obtain a new method for computing the growth function of the Artin monoid of
type $A_{n-1}$ with respect to the simple elements (permutation braids) as
generators. Instead of matrices of size $2^{n-1}\times 2^{n-1}$, we use
matrices of size $p(n)\times p(n)$, where $p(n)$ is the number of partitions of
$n$.

We modify tools introduced by Daniel Daly and Petr Vojtechovsky in order to
count, for any odd prime q, the number of nilpotent loops of order 2q up to
isotopy, instead of isomorphy.

Given a group automorphism $\phi:\Gamma\lr \Gamma$, one has an action of
$\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$.
The orbits of this action are called $\phi$-conjugacy classes. One says that
$\Gamma$ has the $R_\infty$-property if there are infinitely many
$\phi$-conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this
paper we show that any irreducible lattice in a connected semi simple Lie group
having finite centre and rank at least 2 has the $R_\infty$-property.

We consider the class of finitely generated groups which have a normal form
computable in logspace. We prove that the class of such groups is closed under
finite extensions, finite index subgroups, direct products, wreath products,
and also certain free products, and includes the solvable Baumslag-Solitar
groups, as well as non-residually finite (and hence non-linear) examples. We
define a group to be logspace embeddable if it embeds in a group with normal
forms computable in logspace. We prove that finitely generated nilpotent groups
are logspace embeddable.

A subgroup Q is commensurated in a group G if each G conjugate of Q
intersects Q in a group that has finite index in both Q and the conjugate. So
commensurated subgroups are similar to normal subgroups. Semistability and
simple connectivity at infinity are geometric asymptotic properties of finitely
presented groups. In this paper we generalize several of the classic
semistability and simple connectivity at infinity results for finitely
presented groups.

If $G$ is a finite primitive complex reflection group, all reflection
subgroups of $G$ and their inclusions are determined up to conjugacy. As a
consequence, it is shown that if the rank of $G$ is $n$ and if $G$ can be
generated by $n$ reflections, then for every set $R$ of $n$ reflections which
generate $G$, every subset of $R$ generates a parabolic subgroup of $G$.

If $G$ is a group of permutations of a set $\Omega$ and $\alpha \in \Omega$,
then the {\em $\alpha$-suborbits} of $G$ are the orbits of the stabilizer
$G_\alpha$ on $\Omega$. The cardinality of an $\alpha$-suborbit is called a
{\em subdegree} of $G$. If the only $G$-invariant equivalence classes on
$\Omega$ are the trivial and universal relations, then $G$ is said to be a {\em
primitive} group of permutations of $\Omega$.

In this paper we determine the structure of all primitive permutation groups
whose subdegrees are bounded above by a finite cardinal number.

We construct words with small image in a given finite alternating or
unimodular group. This shows that word width in these groups is unbounded in
general.

The main theorem of this paper classifies the quasi-geodesics in a Coxeter
group that are tracked by geodesics. As corollaries, we show that if a Coxeter
group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are
tracked by Cayley graph geodesics, all special subgroups of the Coxeter group
are quasi-convex in X, and in Cayley graphs for Coxeter groups, elements of
infinite order are tracked by geodesics.

We present two results on expansion of Cayley graphs. The first result
settles a conjecture made by DeVos and Mohar. Specifically, we prove that for
any positive constant $c$ there exists a finite connected subset $A$ of the
Cayley graph of $\mathbb{Z}^2$ such that $\frac{|\partial A|}{|A|}<
\frac{c}{depth(A)}$. This yields that there can be no universal bound for
$\frac{|\partial A|depth(A)}{|A|}$ for subsets of either infinite or finite
vertex transitive graphs.

This paper studies classes in Moore's measurable cohomology theory for
locally compact groups and Polish modules. An elementary dimension-shifting
argument is used to show that all such classes have representatives with
considerable extra topological structure beyond measurability. Based on this
idea, for certain target modules one can also construct a direct comparison map
with a different cohomology theory for topological groups defined by Segal, and
show that this map is an isomorphism.

We associate to each finite presentation of a group G a compact CW-complex
that is a 3-manifold in the complement of a point, and whose fundamental group
is isomorphic to G. We use this complex to define a notion of genus for G and
give examples, and also define a notion of `closed group'. A group has genus 0
if and only if it is the fundamental group of a compact orientable 3-manifold.

A semigroup is \emph{amiable} if there is exactly one idempotent in each
$\mathcal{R}^*$-class and in each $\mathcal{L}^*$-class. A semigroup is
\emph{adequate} if it is amiable and if its idempotents commute. We
characterize adequate semigroups by showing that they are precisely those
amiable semigroups which do not contain isomorphic copies of two particular
nonadequate semigroups as subsemigroups.

A structure of a complete lattice (in the sense of a poset) is defined on the
underlying set of the orhtogonal group of a real Euclidean space, by a
construction analogous to that of the weak order of a Coxeter system in terms
of its root system. This gives rise to a complte rootoid in the sense of Dyer,
with the orthogonal group as underlying group.

We study the Dehn function at infinity in the mapping class group, finding a
polynomial upper bound of degree four. This is the same upper bound that holds
for arbitrary right-angled Artin groups.

This note describes a unified approach to several superrigidity results, old
and new, concerning representations of lattices into simple algebraic groups
over local fields. For an arbitrary group $\Gamma$ and a $\Gamma$-boundary $B$
we associate certain generalized Weyl group $W_{\Gamma,B}$ and show that any
representation with a Zariski dense unbounded image in a simple algebraic
group, $\rho:\Gamma\to \mathbf{H}$, defines a special homomorphism
$W_{\Gamma,B}\to {\rm Weyl}(\mathbf{H})$. This general fact allows to deduce
the aforementioned superrigidity results.

We construct a class of finitely presented groups where the isomorphism
problem is solvable but the commensurability problem is unsolvable. Conversely,
we construct a class of finitely presented groups within which the
commensurability problem is solvable but the isomorphism problem is unsolvable.
These are first examples of such a contrastive complexity behaviour with
respect to the isomorphism problem.

We study abstract finite groups with the property, called property $\hat{s}$,
that all of their subrepresentations have submultiplicative spectrum. Such
groups are necessarily nilpotent and we focus on $p$-groups. $p$-groups with
property $\hat{s}$ are regular. Hence, a 2-group has property $\hat{s}$ if and
only if it is commutative. For an odd prime $p$, all $p$-abelian groups have
property $\hat{s}$, in particular all groups of exponent $p$ have it. We show
that a 3-group or a metabelian $p$-group ($p \ge 5$) has property $\hat{s}$ if
and only if it is V-regular.

Let G be a complete Kac-Moody group over a finite field. It is known that G
possesses a BN-pair structure, all of whose parabolic subgroups are open in G.
We show that, conversely, every open subgroup of G has finite index in some
parabolic subgroup. The proof uses some new results on parabolic closures in
Coxeter groups. In particular, we give conditions ensuring that the parabolic
closure of the product of two elements in a Coxeter group contains the
respective parabolic closures of those elements.

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we
calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n.
If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the
determination of the order of $G_n$ is split into three cases, according as e
is non-symmetric, non-constant symmetric, or constant. The formulas that we
obtain only depend on p, n, and the rank of the circulant matrix whose first
row is e.

A ballean is a set endowed with some family of its subsets which are called
the balls. We postulate the properties of the family of balls in such a way
that the balleans can be considered as the asymptotic counterparts of the
uniform topological spaces. The isomorphisms in the category of balleans are
called asymorphisms. Every metric space can be considered as a ballean. The
ultrametric spaces are prototypes for the cellular balleans. We prove some
general theorem about decomposition of a homogeneous cellular ballean in a
direct product of a pointed family of sets.

A result of Ben-Or, Coppersmith, Luby and Rubinfeld on testing whether a map
be two groups is close to a homomorphism implies a tight lower bound on the
distance between the multiplication tables of two non-isomorphic groups.

We find finite presentations for the automorphism group of the Artin pure
braid group and the automorphism group of the pure braid group associated to
the full monomial group.

We prove the Girth Alternative for finitely generated subgroups of PL_o(I).
We also prove that a finitely generated subgroup of Homeo(I) which is
sufficiently rich with hyperbolic-like elements has infinite girth.

We consider Thompson's groups from the perspective of mapping class groups of
surfaces of infinite type. This point of view leads us to the braided Thompson
groups, which are extensions of Thompson's groups by infinite (spherical) braid
groups. We will outline the main features of these groups and some applications
to the quantization of Teichm\"uller spaces. The chapter provides an
introduction to the subject with an emphasis on some of the authors results.

Let K be a number field and let A be its ring of integers. Let G be a
connected, noncommutative, absolutely almost simple algebraic K-group. If the
K-rank of G equals 2, then G(A[t]) is not finitely presented.

In this paper we study numerical semigroups generated by three elements. We
give a characterization of pseudo-symmetric numerical semigroups. Also, we will
give a simple algorithm to get all the pseudo-symmetric numerical semigroups
with give Frobenius number.

For $N\geq 4$, we show that there exist automorphisms of the free group $F_N$
which have a parabolic orbit in $\partial F_N$. In fact, we exhibit a
technology for producing infinitely many such examples.

We prove that every finitely generated group with recursive aspherical
presentation embeds into a group with finite aspherical presentation. This and
several known facts about groups and manifolds imply that there exists a
4-dimensional closed aspherical manifold $M$ such that the fundamental group
$\pi_1(M)$ coarsely contains an expander, and so it has infinite asymptotic
dimension, is not coarsely embeddable into a Hilbert space, and does not
satisfy the Baum-Connes conjecture with coefficients. Closed aspherical
manifolds with any of these properties were previously unknown.

We study symmetric random walks on finitely generated groups of
orientation-preserving homeomorphisms of the real line. We establish an
oscillation property for the induced Markov chain on the line that implies a
weak form of recurrence. Except for a few special cases, which can be treated
separately, we prove a property of "global stability at a finite distance":
roughly speaking, there exists a compact interval such that any two
trajectories get closer and closer whenever one of them returns to the compact
interval.

The Cayley-Dickson loop C_n is the multiplicative closure of basic elements
of the algebra constructed by n applications of the Cayley-Dickson doubling
process (the first few examples of such algebras are real numbers, complex
numbers, quaternions, octonions, sedenions). We discuss properties of the
Cayley-Dickson loops, show that these loops are Hamiltonian and describe the
structure of their automorphism groups.

Consider a group word w in n letters. For a compact group G, w induces a map
G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar
measure on G^n. We associate to each word w a 2-dimensional cell complex X(w)
and prove in Theorem 2.5 that {\mu}_w is determined by the topology of X(w).
The proof makes use of non-abelian cohomology and Nielsen's classification of
automorphisms of free groups [Nie24].

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers
in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is
quasi-isometric to a tree. This implies that a group $G$ acting properly and
cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane
stabilizers. Another application affirms Sageev's finite hyperplane coloring
conjecture for uniformly locally finite CAT(0) cube complexes, generalizing
Sageev's results in the $\delta$-hyperbolic case.

An arbitrary homomorphism between groups is nonincreasing for stable
commutator length, and there are infinitely many (injective) homomorphisms
between free groups which strictly decrease the stable commutator length of
some elements. However, we show in this paper that a random homomorphism
between free groups is almost surely an isometry for stable commutator length
for every element; in particular, the unit ball in the scl norm of a free group
admits an enormous number of exotic isometries.

We construct a family of finite 2-complexes whose universal covers are CAT(0)
and have polynomial divergence of desired degree. This answers a question of
Gersten, namely whether such CAT(0) complexes exist.

We construct the first example of a coarsely non-amenable (= without Guoliang
Yu's property A) metric space with bounded geometry which coarsely embeds into
a Hilbert space.

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any
two finitely generated and non-conjugate subgroups of G remain non-conjugate in
some finite quotient of G. We prove that the free groups and the fundamental
groups of finite trees of finite groups with some normalizer condition are SCS.
We also introduce the subgroup into-conjugacy separability property and prove
that the above groups have this property too.

Let k be an algebraically closed field of characteristic 2, and let W be the
ring of infinite Witt vectors over k. Suppose G is a finite group and B is a
block of kG with a dihedral defect group D such that there are precisely two
isomorphism classes of simple B-modules. We determine the universal deformation
ring R(G,V) for every finitely generated kG-module V which belongs to B and
whose stable endomorphism ring is isomorphic to k.

We survey results about computational complexity of the word problem in
groups, Dehn functions of groups and related problems.

Let M be a commutative monoid. We provide an explicit first-order formular
that defines the variety generated by M in the lattice of commutative semigroup
varieties.

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show
that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly
amenable. We will study this problem under generalization, that is, if
$(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$
is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from
$A^{(n)}$ into $A^{(n)}$.

In this work we study the asymptotic traffic behavior in Gromov's hyperbolic
spaces when the traffic decays exponentially with the distance. We prove that
under general conditions, there exist a phase transition between local and
global traffic.

Consider a relatively hyperbolic group G. We prove that if G is finitely
presented, so are its parabolic subgroups. Moreover, a presentation of the
parabolic subgroups can be found algorithmically from a presentation of G, a
solution of its word problem, and generating sets of the parabolic subgroups.
We also give an algorithm that finds parabolic subgroups in a given recursively
enumerable class of groups.

We exhibit an infinitely presented 4-soluble group with Cantor-Bendixson rank
one, and consequently with no minimal presentation. Then we study the class of
infinitely presented metabelian groups lying in the condensation part of the
space of marked groups.

We develop some new topological tools to study maximal subgroups of free
idempotent generated semigroups. As an application, we show that the rank 1
component of the free idempotent generated semigroup of the biordered set of a
full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal
subgroup isomorphic to the multiplicative subgroup of Q.

For a countable group G and a multiplier c on G with values in the circle, we
study the property of G having a unitary projective c-representation which is
both irreducible and projectively faithful. We show that this property is
equivalent to G being the quotient of an appropriate group by its centre. A
criterion is given in terms of the minisocle of G. Several examples are
described to show the existence of various behaviours.

Let \Gamma be a non-cocompact lattice on a locally finite regular
right-angled building X. We prove that if \Gamma has a strict fundamental
domain then \Gamma is not finitely generated. We use the separation properties
of subcomplexes of X called tree-walls.

Based on an idea of Y.P\'eresse and some results of Maltcev, Mitchell and
Ru\v{s}kuc, we present sufficient conditions under which the endomorphism
monoid of an ultrahomogeneous first-order structure has the Bergman property.
This property has played a prominent role both in the theory of infinite
permutation groups and, more recently, in semigroup theory.

In this paper, we prove a limit set intersection theorem in relatively
hyperbolic groups. We also show that a nonparabolic relatively quasiconvex
subgroup cannot contain a proper conjugate of itself. Several well-known
results on limit sets of geometrically finite Kleinian groups are derived in
relatively hyperbolic groups. Lastly, we establish the dynamical quasiconvexity
for undistorted subgroups of finitely generated groups with nontrivial Floyd
boundary.

We present a polynomial time Monte-Carlo algorithm for finite simple black
box classical groups of odd characteristic which constructs all root
${\rm{SL}}_2(q)$-subgroups associated with the nodes of the extended Dynkin
diagram of the corresponding algebraic group.

We exhibit a 6-element semigroup that has no finite identity basis but
nevertheless generates a variety whose finite membership problem admits a
polynomial algorithm.

We study stable W-length in groups, especially for W equal to the n-fold
commutator gamma_n:=[x_1,[x_2, . . . [x_{n-1},x_n]] . . . ]. We prove that in
any perfect group, for any n at least 2 and any element g, the stable
commutator length of g is at least as big as 2^{2-n} times the stable
gamma_n-length of g. We also establish analogues of Bavard duality for words
gamma_n and for beta_2:=[[x,y],[z,w]]. Our proofs make use of geometric
properties of the asymptotic cones of verbal subgroups with respect to
bi-invariant metrics.

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup
of a subgroup $V$. Suppose $\phi \in \ibr G$. If either $|G|$ is odd or $p =
2$, we prove that the number of Brauer characters of $H$ inducing $\phi$ with
vertex $Q$ is at most $|\norm GQ: \norm VQ|$.

A Jordan loop is a commutative loop satisfying the Jordan identity $(x^2 y) x
= x^2 (y x)$. We establish several identities involving powers in Jordan loops
and show that there is no nonassociative Jordan loop of order $9$.

We consider the structure of finite $p$-groups $G$ having precisely three
characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. The structure of $G$
varies markedly depending on whether $G$ has exponent $p$ or $p^2$, and, in
both cases, the study of such groups raises deep problems in representation
theory. We present classification theorems for 3- and 4-generator groups, and
we also study the existence of such $r$-generator groups with exponent $p^2$
for various values of $r$.

In this paper, we show that any Coxeter graph which defines a higher rank
Coxeter group must have disjoint induced subgraphs each of which defines a
hyperbolic or higher rank Coxeter group. We then use this result to demonstrate
several classes of Coxeter graphs which define hyperbolic Coxeter groups.

This paper is devoted to the investigation of the property of order
separability for HNN extensions and free products with commutative subgroups.
Particularly it was proven that HNN extension of a free group with maximal
connected cyclic subgroups is 2-order separable.

In this paper we introduce the concept of weighted deficiency for abstract
and pro-$p$ groups and study groups of positive weighted deficiency which
generalize Golod-Shafarevich groups. In order to study weighted deficiency we
introduce weighted versions of the notions of rank for groups and index for
subgroups and establish weighted analogues of several classical results in
combinatorial group theory, including the Schreier index formula. Two main
applications of groups of positive weighted deficiency are given.

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian
group $K$, the generalized shift $\sigma_\lambda$ of the direct sum
$\bigoplus_\Gamma K$ is the endomorphism defined by
$(x_i)_{i\in\Gamma}\mapsto(x_{\lambda(i)})_{i\in\Gamma}$. In this paper we
analyze and compute the algebraic entropy of a generalized shift, which turns
out to depend on the cardinality of $K$, but mainly on the function $\lambda$.
We give many examples showing that the generalized shifts provide a very useful
universal tool for producing counter-examples.

We extend to endomorphism of arbitrary abelian groups the definition of the
algebraic entropy h given by Peters for automorphisms and we study the
properties of h. In particular, we prove the Addition Theorem for h and we
obtain a Uniqueness Theorem for h in the category of all abelian groups and
their endomorphisms. The third of our main results is the Bridge Theorem
connecting the algebraic entropy and the topological entropy by the Pontryagin
duality.

We observe that a Polish group $G$ is amenable if and only if every
continuous action of $G$ on the Hilbert cube admits an invariant probability
measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that
actions on the Cantor space can be used to detect amenability and extreme
amenability of Polish non-archimedean groups as well as amenability at infinity
of discrete countable groups. As corollary, the latter property can also be
tested by actions on the Hilbert cube. These results generalise a criterion due
to Giordano and de la Harpe.

We analyze the Sibert et al. group-based (Feige-Fiat-Shamir type)
authentication protocol and show that the protocol is not computationally
zero-knowledge. In addition, we provide experimental evidence that our approach
is practical and can succeed even for groups with no efficiently computable
length function such as braid groups. The novelty of this work is that we are
not attacking the protocol by trying to solve an underlying complex algebraic
problem, namely, the conjugacy search problem, but use a probabilistic
approach, instead.

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it
generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G)
with respect to the generating set pi_q(S) form a family of expanders, where
pi_q is the projection map Z->Z/qZ.

Motivated by algorithmic problems from combinatorial group theory we study
computational properties of integers equipped with binary operations +, -, z =
x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice
that in this case very large numbers, which are obtained as n towers of
exponentiation in the base 2 can be realized as n applications of the operation
x2^y, so working with such numbers given in the usual binary expansions
requires super exponential space.

We prove that the Higman-Thompson groups $G_{n,r}^+$ and $G_{m,s}^+$ are
isomorphic if and only if $m=n$ and $\mbox{gcd}(n-1,r)=\mbox{gcd}(n-1,s)$.

A notion of degeneration of elements in groups is introduced. It is used to
parametrize the orbits in a finite abelian group under its full automorphism
group by a finite distributive lattice. A pictorial description of this lattice
leads to an intuitive self-contained exposition of some of the basic facts
concerning these orbits, including their enumeration. Given a partition
$\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type
$\lambda$ is found to be independent of p.

We study profinite actions of residually finite groups in terms of weak
containment. We show that two strongly ergodic profinite actions of a group are
weakly equivalent if and only if they are isomorphic. This allows us to
construct continuum many pairwise weakly inequivalent free actions of a large
class of groups, including free groups and linear groups with property (T). We
also prove that for chains of subgroups of finite index, Lubotzky's property
($\tau$) is inherited when taking the intersection with a fixed subgroup of
finite index.

We prove that an irreducible lattice acting on a product of two or more
locally finite, biregular trees is finitely generated.

According to Markov, a subset of an abelian group G of the form {x in G:
nx=a}, for some integer n and some element a of G, is an elementary algebraic
set; finite unions of elementary algebraic sets are called algebraic sets. We
prove that a subset of an abelian group G is algebraic if and only if it is
closed in every precompact (=totally bounded) Hausdorff group topology on G.
The family of all algebraic subsets of an abelian group G forms the family of
closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or
verbal, topology of G.

We study equations in groups G with unique m-th roots for each positive
integer m. A word equation in two letters is an expression of the form w(X,A) =
B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as
fixed coefficients, and X in G as the unknown. Certain word equations, such as
XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do
not. We obtain the first known infinite families of word equations not solvable
by radicals, and conjecture a complete classification.

There are 2 parts. To part one. P. Broussous and S. Stevens studied maps
between enlarged Bruhat-Tits buildings to construct types for p-adic unitary
groups. They needed maps which respect the Moy-Prasad filtrations. That
property is called (CLF), i.e. compatibility with the Lie algebra filtrations.
We generalise their results on CLF-maps. Let k_0 be a p-adic field of
characteristic not two. We consider G:={\bf U}(h) defined over k_0 with a
signed hermitian form h. Let H be the centraliser of a semisimple k_0-rational
Lie algebra element of G.

We characterize permutational wreath products with Property (FA). For
instance, the standard wreath product A wr B of two nontrivial countable groups
A,B, has Property (FA) if and only if B has Property (FA) and A is a finitely
generated group with finite abelianisation. We also prove an analogous result
for hereditary Property (FA). On the other hand, we prove that many wreath
products with hereditary Property (FA) are not quotients of finitely presented
groups with the same property.

We introduce a criterion for the completeness of ring approximations of
modular cohomology rings of finite non prime power groups, and provide an
example for which it performs substantially better than previously established
criteria.

We show that topological amenability of an action of a countable discrete
group on a compact space is equivalent to the existence of an invariant mean
for the action. We prove also that this is equivalent to vanishing of bounded
cohomology for a class of Banach G-modules associated to the action, as well as
to vanishing of a specific cohomology class. In the case when the compact space
is a point our result reduces to a classic theorem of B.E. Johnson
characterising amenability of groups.

Let F be an infinitely generated free group and R a fully invariant subgroup
of F such that (a) R is contained in the commutator subgroup F' of F and (b)
the quotient group F/R is residually torsion-free nilpotent. Then the
automorphism group Aut(F/R') of the group F/R' is complete. In particular, the
automorphism group of any infinitely generated free solvalbe group of derived
length at least two is complete.

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is
generated by at most d elements, and such that p is the largest prime dividing
|G|. We show that G has a non-nilpotent image G/N, such that N is
characteristic and of index bounded by a function of d and p. This result will
be used to prove that the index of the Frattini subgroup of G is bounded in
terms of d and p. Upper bounds will be given explicitly for soluble groups.

We study hereditary properties of the class $\mathcal{A}$ defined by
Glasner-Monod of countable groups admitting an amenable, transitive and
faithful action. We consider mainly the case of amalgamated free products, and
we show in particular that the double of amenable groups and the amalgamated
free products of two amenable groups over a finite subgroup are contained in
$\mathcal{A}$.

It is shown that for graph groups (right-angled Artin groups) the conjugacy
problem as well as a restricted version of the simultaneous conjugacy problem
can be solved in polynomial time even if input words are represented in a
compressed form. As a consequence it follows that the word problem for the
outer automorphism group of a graph group can be solved in polynomial time.

We study the generalization of the notion of order separability for free
groups.

The author in $($On the order of Schur multiplier of non-abelian $p$-groups.
J. Algebra (2009).322: 4479--4482$)$ showed that for any $p$-group $G$ of order
$p^n$ there exists a nonnegative integer $s(G)$ such that the order of Schur
multiplier of $G$ is equal to $p^{\f{1}{2}(n-1)(n-2)+1-s(G)}$. Furthermore, he
characterized the structure of all non-abelian $p$-groups $G$ when $s(G)=0$.
The present paper is devoted to characterization of all $p$-groups when
$s(G)=2$.

Let $W$ be a complex reflection group and $B$ and $P$ be respectively the
braid group and the pure braid group of $W$.

The aim of this article is to give a nice description of the following short
exact sequence as an element of $H^2(W,P^\textrm{ab})$.

$$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] &
1}$$

The commuting graph of a group G, denoted by Gamma(G), is the simple
undirected graph whose vertices are the non-central elements of G and two
distinct vertices are adjacent if and only if they commute. Let Z_m be the
commutative ring of equivalence classes of integers modulo m. In this paper we
investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m)
to contribute to the conjecture that there is a universal upper bound on
diam(Gamma(G)) for any finite group G when Gamma(G) is connected.

We completely determine all lower-modular elements of the lattice of all
semigroup varieties. As a corollary, we show that a lower-modular element of
this lattice is modular.

An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings
are automorphisms. Every element of a commutative A-loop generates a group, and
$(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and
$p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element
of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of
a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order,
it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$.

We prove that an Artin-Tits group of affine type C has a (dual) Garside
structure which we obtain by viewing it as the group of fixed points under an
involution in an Artin-Tits group of affine type A.

We consider the class of those Coxeter groups for which removing from the
Cayley graph any tubular neighbourhood of any wall leaves exactly two connected
components. We call these Coxeter groups bipolar. They include both the
virtually Poincare duality Coxeter groups and the infinite irreducible
2-spherical ones. We show in a geometric way that a bipolar Coxeter group
admits a unique conjugacy class of Coxeter generating sets. Moreover, we
provide a characterisation of bipolar Coxeter groups in terms of the associated
Coxeter diagram.

This is a survey of way that the sizes of conjugacy classes influence the
structure of finite groups

Let $T$ be an $\mathbb{R}$-tree, equipped with a very small action of the
rank $n$ free group $F_n$, and let $H \leq F_n$ be finitely-generated. We
consider the case where $T$ is indecomposable--this is a strong mixing property
introduced by Guirardel. In this case, we show that the action of $H$ on its
minimal invarinat subtree $T_H$ is indiscrete if and only if $H$ is finite
index in $F_n$.

Broadly speaking, a finiteness property of groups is any generalisation of
the property of having finite order. A large part of infinite group theory is
concerned with finiteness properties and the relationships between them.
Profinite groups are an important case of this, being compact topological
groups that possess an intimate connection with their finite images. This
thesis investigates the relationship between several finiteness properties that
a profinite group may have, with consequences for the structure of finite and
profinite groups.

It is well known that the triviality problem for finitely presented groups is
unsolvable. We ask the question of whether there exists a general procedure to
produce a non-trivial element from a finite presentation of a non-trivial
group. If not, then this would resolve an open problem by J. Wiegold: `Is every
finitely generated perfect group the normal closure of one element?' We prove a
weakened version of our question; there is no general procedure to pick a
non-trivial generator from a finite presentation of a non-trivial group.

For a group $H$ and a subset $X$ of $H$, we let ${}^HX$ denote the set
$\{hxh^{-1} \mid h \in H, x \in X\}$, and when $X$ is a free-generating set of
$H$, we say that the set ${}^HX$ is a Whitehead subset of $H$.

For a group $F$ and an element $r$ of $F$, we say that $r$ is Cohen-Lyndon
aspherical in $F$ if ${}^F\{r\}$ is a Whitehead subset of the subgroup of $F$
that is generated by ${}^F\{r\}$.

In 1963, D. E. Cohen and R. C. Lyndon independently showed that in each free
group each non-trivial element is Cohen-Lyndon aspherical.

Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose
that |A|<p^{n+1-\delta}, delta>0. Then

|A A A|>>|A|^{1+\epsilon}, where epsilon>0 and the implied constant depend
only on n and delta.

We prove for many groups considered in classical mathematics (Chevalley
groups over infinite fields, connected perfect linear algebraic groups,
infinite permutation and infinite dimensional general linear groups), a model
theoretical phenomenon called absolutely connectedness. Namely, G is absolutely
connected if for an arbitrary first order structure on G, working in a
saturated extension, G does not have any proper definable, type definable or
invariant under structure automorphisms subgroups of bounded index i.e.
G=G^0=G^00=G^{\infty}.

We associate with every Renner monoid $R$ a \emph{generic Hecke algebra}
$\H(R)$ over $\mathbb{Z}[q]$ which is a deformation of the monoid
$\mathbb{Z}$-algebra of $R$. If $M$ is a finite reductive monoid with Borel
subgroup $B$ and associated Renner monoid $R$, then we obtain the associated
Iwahori-Hecke algebra $\H(M,B)$ by specialising $q$ in $\H(R)$ and tensoring by
$\mathbb{C}$ over $\mathbb{Z}$, as in the classical case of finite algebraic
groups. This answers positively to a long-standing question of L. Solomon.

In the given paper we prove that every automorphism of a Chevalley group of
type $A_l$, $D_l$, or $E_l$, $l\geqslant 3$, over a commutative local ring
without 1/2 is standard, i. e., it is a composition of ring, inner, central and
graph automorphisms.

This paper provides a realization of all classical and most exceptional
finite groups of Lie type as Galois groups over function fields over F_q and
derives explicit additive polynomials for the extensions. Our unified approach
is based on results of Matzat which give bounds for Galois groups of Frobenius
modules and uses the structure and representation theory of the corresponding
connected linear algebraic groups.

We examine groups whose resonance varieties, characteristic varieties and
Sigma-invariants have a natural arithmetic group symmetry, and we explore
implications on various finiteness properties of subgroups. We compute
resonance varieties, characteristic varieties and Alexander polynomials of
Torelli groups, and we show that all subgroups containing the Johnson kernel
have finite first Betti number, when the genus is at least four.

Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with
uniformly bounded number of generators. We prove that the infinite iterated
permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely
generated if and only if the profinite abelian group $G_1/G'_1\oplus
G_2/G'_2\oplus...$ is topologically finitely generated. As a corollary, for a
finite transitive group $G$ the minimal number of generators of the wreath
power $G\wr...\wr G\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows
linearly if $G$ is non-perfect.

Automorphism groups of locally finite trees provide a large class of examples
of simple totally disconnected locally compact groups. It is desirable to
understand the connections between the global and local structure of such a
group. Topologically, the local structure is given by the commensurability
class of a vertex stabiliser; on the other hand, the action on the tree
suggests that the local structure should correspond to the local action of a
stabiliser of a vertex on its neighbours.

We explain, following Gromov, how to produce uniform isometric actions of
groups starting from isometric actions without fixed point, using common
ultralimits techniques. This gives in particular a simple proof of a result by
Shalom: Kazhdan's property (T) defines an open subset in the space of marked
finitely generated groups.

We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A
is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon
is an absolute positive real number, or AAA=G.

As a corollary we get that the diameter of any Cayley graph of G is
Poly-Logarithmic in |G|.

In this article we relate word and subgroup growth to certain functions that
arise in the quantification of residual finiteness. One consequence of this
endeavor is a pair of results that equate the nilpotency of a finitely
generated group with the asymptotic behavior of these functions. The second
half of this article investigates the asymptotic behavior of two of these
functions. Our main result in this arena resolves a question of Bogopolski from
the Kourovka notebook concerning lower bounds of one of these functions for
nonabelian free groups.

A continuous equivariant map from the Floyd boundary of a finitely generated
relatively hyperbolic group (RHG for short) to its Bowditch boundary is
constructed. Such a map is unique unless the group is two-ended. We consider a
RHG as a group acting on a compactum discontinuously on triples and cocompactly
on pairs. We describe and make use a rather general construction of "attractor
sum" of two actions of a locally compact group.

Lyashko-Looijenga morphisms (LL) are non-Galois finite extension of
polynomial rings, occuring in the geometry of well-generated reflection groups
and their associated braid groups. Bessis suggested a program to study them as
"virtual reflection groups" (analogies with invariant theory, but without a
true group action). A first step involves Jacobians and discriminants
associated to LL: our main result is that they indeed behave as expected. The
proof uses commmutative algebra and a combinatorial description of the fibers
of LL. As byproducts, we recover a formula stated by K.

Let $G$ be a finite $p$-group of order $p^n$. It is known that
$|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$ and $t(G)\geq 0$. The structure of
$G$ characterized when $t(G)\leq 4$ in \cite{be,el,ni,sa,zh}. The structure
description of $G$ is determined in this paper for $t(G)=5$.

In this survey we show how well known results about the Word Problem for
finite group presentations can be generalized to the Word Problem and other
decision problems for non-necessarily finite monoid and group presentations.
This is done by introducing functions playing the same role of the Dehn
function for the given decision problem and by finding the Tietze
transformations that leave this function invariant. This survey presents some
original ideas and points of view.

Let $G$ be a finite group, $F$ a field, and $V$ a finite dimensional
$FG$-module such that $G$ has no trivial composition factor on $V$. Then the
arithmetic average dimension of the fixed point spaces of elements of $G$ on
$V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the
order of $G$. This answers and generalizes a 1966 conjecture of Neumann which
also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in
The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a
recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret\'o.

We give a group theoretic characterization of geodesics with superlinear
divergence in the Cayley graph of a right-angled Artin group A(G) with
connected defining graph G. We use this to determine when two points in an
asymptotic cone of A(G) are separated by a cut-point. As an application, we
show that if G does not decompose as the join of two subgraphs, then A(G) has
an infinite-dimensional space of non-trivial quasimorphisms. By the work of
Burger and Monod, this leads to a superrigidity theorem for homomorphisms from
lattices into right-angled Artin groups.

Let S be a fixed symmetric finite subset of SL_d(O_K) that generates a
Zariski dense subgroup of SL_d(O_K) when we consider it as an algebraic group
over Q by restriction of scalars. We prove that the Cayley graphs of
SL_d(O_K/I) with respect to the projections of S is an expander family if I
ranges over square-free ideals of O_K if d=2 and K is an arbitrary numberfield,
or if d=3 and K=Q.

This is a survey of two papers joint with A. Borisov and a paper joint with
I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews
2009", Bath (August 2009). We prove that almost all 1-related groups with at
least 3 generators are virtually residually (finite p-)groups for almost all
primes p, and coherent.

Recall that a homomorphism of $R$-modules $\pi: G\to H$ is called a {\it
cellular cover} over $H$ if $\pi$ induces an isomorphism $\pi_*:
\Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\varphi)= \pi \varphi$ for each
$\varphi \in \Hom_R(G,G)$ (where maps are acting on the left). In this paper we
show that every cotorsion-free module $K$ of finite rank can be realized as the
kernel of a cellular cover of some cotorsion-free module of rank 2. In
particular, every free abelian group of any finite rank appears then as the
kernel of a cellular cover of a cotorsion-free abelian group of rank 2.

We describe the structure, including composition factors and submodule
lattices, of cross-characteristic permutation modules for the natural actions
of the orthogonal groups $O_{m}^{\pm}(3)$ with $m\geq6$ on nonsingular points
of their standard modules. These actions together with those studied in
\cite{HN} are all examples of primitive rank 3 actions of finite classical
groups on nonsingular points.

We say that a subset $S\subseteq F_N$ is \emph{spectrally rigid} if whenever
$T_1, T_2\in cv_N$ are points of the (unprojectivized) Outer space such that
$||g||_{T_1}=||g||_{T_2}$ for every $g\in S$ then $T_1=T_2$ in $\cvn$. It is
well-known that $F_N$ itself is spectrally rigid; it also follows from the
result of Smillie and Vogtmann that there does not exist a finite spectrally
rigid subset of $F_N$.

While lattices in semi-simple Lie groups are studied very well, only little
is known about discrete subgroups of infinite covolume. The main class of
examples are Schottky groups. Here we investigate some new examples.

In previous work, the first author established a natural bijection between
minimal subshifts and maximal regular J-classes of free profinite semigroups.
In this paper, the Schutzenberger groups of such J-classes are investigated in
particular in respect to a conjecture proposed by the first author concerning
their profinite presentation. The conjecture is established for several types
of minimal subshifts associated with substitutions.

This paper addresses various questions about pairs of similarity classes of
matrices which contain commuting elements. In the case of matrices over finite
fields, we show that the problem of determining such pairs reduces to a
question about nilpotent classes; this reduction makes use of class types in
the sense of Steinberg and Green. We investigate the set of scalars that arise
as determinants of elements of the centralizer algebra of a matrix, providing a
complete description of this set in terms of the class type of the matrix.

Using Lipschitz distance on Outer space we give another proof of the train
track theorem.

We show that for some negatively curved solvable Lie groups, all self
quasiisometries are almost isometries. We prove this by showing that all self
quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are
bilipschitz with respect to the visual metric. We also define parabolic visual
metrics on the ideal boundary of Gromov hyperbolic spaces and relate them to
visual metrics.

Let k be a field of characteristic p>0, and G be a finite group. The first
result of this paper is an explicit formula for the determinant of the Cartan
matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula
for the rank of the Cartan matrix of the cohomological Mackey algebra comu_k(G)
of G over k, and a characterization of the groups G for which this matrix is
non singular.

We reduce the classification of finite subgroups in compact Lie groups to
that of quasi-simple ones, prove the number of conjugacy classes is finite and
each cojugacy class is Zariski closed in mapping space, and classify ``strongly
controlling fusions" symmetric pairs.

We summarize the main known results involving subword reversing, a method of
semigroup theory for constructing van Kampen diagrams by referring to a
preferred direction. In good cases, the method provides a powerful tool for
investigating presented (semi)groups. In particular, it leads to cancellativity
and embeddability criteria for monoids and to efficient solutions for the word
problem of monoids and groups of fractions.

We prove that, for r=3 and 4, the minimal nonabelian finite quotient of the
outer automorphism group Out F_r of a free group of rank r is the linear group
PSL_r(Z_2) (this might remain true, however, for arbitrary rank r > 2).

Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a
prime $p$ in $\pi(G)$, the set of all prime divisors of the order of $G$. The
automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define
the Sylow graph $\Gamma_A(G)$ of the group $G$, with set of vertices $\pi(G)$,
as follows: Two vertices $p,q\in\pi(G)$ form an edge of $\Gamma_A(G)$ if either
$q\in\pi(A_p(G))$ or $p\in \pi(A_q(G))$. The following result is obtained:

Theorem: Let $G$ be a finite almost simple group. Then the graph
$\Gamma_A(G)$ is connected and has diameter at most 5.

We consider the structure of a finite groups having a normal series whose
factors have bicyclic Sylow subgroups. In particular, we investigated groups of
odd order and $A_4$-free groups with this property. Exact estimations of the
derived length and nilpotent length of such groups are obtained.

We find a lower bound to the size of finite groups detecting a given word in
the free group, more precisely we construct a word w_n of length n in
non-abelian free groups with the property that w_n is the identity on all
finite quotients of size ~ n^{2/3} or less. This improves on a previous result
of Bou-Rabee and McReynolds quantifying the lower bound of the residual
finiteness of free groups.

We prove that non-uniform arithmetic lattices of $SL_2(\mathbb{C})$ and in
particular the Bianchi groups are conjugacy separable. The proof based on
recent deep results of Agol, Long, Reid and Minasyan.

Certain topological properties of the group $\J(\k)$ of all formal
one-variable power series with coefficients in a topological unitary ring $\k$
are considered. We show, in particular, that in the case when $\k=\Q$ the group
$\J(\Q)$ has no continuous bijections into a locally compact group. In the case
when $\k=\Z$ supplied with discrete topology, in spite of the fact that the
group $\J(\Z)$ has continuous bijections into compact groups, it cannot be
embedded into a locally compact group. In the final part of the paper the
compression property for topological groups is considered.

Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by
left- and by right multiplication. This gives rise to a partition of the
complement of T in S, and to each equivalence class of this partition we
naturally associate a relative Schutzenberger group. We show how generating
sets for S may be used to obtain generating sets for T and the Schutzenberger
groups, and vice versa. We also give a method for constructing a presentation
for S from given presentations of T and the Schutzenberger groups.

Let $k$ be an algebraically closed field of characteristic 2, let $G$ be a
finite group with dihedral Sylow 2-subgroups, and let $B$ be the principal
block of $kG$. Assume that there are precisely two isomorphism classes of
simple $B$-modules. The description by Erdmann of the quiver and relations of
the basic algebra of $B$ is usually only given up to a certain parameter $c$
which is either 0 or 1. In this article, we show that $c=0$ if there exists a
central extension $\hat{G}$ of $G$ by a group of order 2 such that the Sylow
2-subgroups of $\hat{G}$ are generalized quaternion.

Shalom characterized property (T) in terms of the vanishing of all reduced
first cohomology. We characterize group pairs having the property that the
restriction map on all first reduced cohomology vanishes. We show that, in a
strong sense, this is inequivalent to relative property (T).