This is a list of problems prepared for the conference ``Problems in Geometric Group Theory" at American Institute of Mathematics in April 2007. Any comments and suggestions are welcome.

We call a finitely generated group coarsely hyperbolic if one of its asymptotic cones is an $\mathbb R$-tree. We characterize coarsely hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants. Using central extensions of coarsely hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptotic cone $C$ has countable but non-trivial fundamental group (in fact $C$ is homeomorphic to the direct product of an $\mathbb R$--tree and a circle, so $\pi_1(C)=\mathbb Z$). We show that the class of coarsely hyperbolic groups contains elementary amenable groups, groups with all proper subgroups cyclic, and torsion groups. This allows us to solve two problems of Drutu and Sapir, and a problem of Kleiner about groups with cut-points in their asymptotic cones. We also construct a finitely generated group whose divergence function is not linear but is arbitrarily close to being linear. This answers a question of Behrstock.

We show that for any non--elementary hyperbolic group $H$ and any finitely presented group $Q$, there exists a short exact sequence $1\to N\to G\to Q\to 1$, where $G$ is a hyperbolic group and $N$ is a quotient group of $H$. As an application we construct a hyperbolic group that has the same $n$--dimensional complex representations as a given finitely generated group, show that adding relations of the form $x^n=1$ to a presentation of a hyperbolic group may drastically change the group even in case $n>> 1$, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier--Wise on outer automorphism groups of Kazhdan groups.

We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the quotient of $G$ by the normal subgroup generated by $g^n$ is large for all but finitely many $n\in \mathbb Z$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $(p_1, p_2, ...)$, there exists an infinite finitely generated periodic group $Q$ with descending normal series $Q=Q_0\rhd Q_1\rhd ... $, such that $\bigcap_i Q_i=\{1\} $ and $Q_{i-1}/Q_i$ is either trivial or abelian of exponent $p_i$.

We study residual properties of relatively hyperbolic groups. In particular, we show that if a group $G$ is non--elementary and hyperbolic relative to a collection of proper subgroups, then $G$ is SQ--universal.

We generalize the small cancellation theory over hyperbolic groups developed by Olshanskii to the case of relatively hyperbolic groups. This allows us to construct infinite finitely generated groups with exactly $n$ conjugacy classes for every $n\ge 2$. In particular, we give the affirmative answer to the well--known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a {\it peripheral filling procedure}, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi _1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings.

The aim of this paper is to show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense. More precisely their left regular representations are isolated from the trivial representation uniformly on finite generating sets. It follows that free Burnside groups are of uniform exponential growth. This answers a question of de la Harpe.

The main purpose of this paper is to explain why, contrary to a prevalent opinion, a public key encryption can be secure against encryption emulation attack by computationally unbounded adversary, with one reservation: a legitimate party decrypts correctly with probability that can be made arbitrarily close to 1, but not equal to 1.

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{ H_\lambda ,\lambda \in \Lambda \} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{ H_\lambda ,\lambda \in \Lambda \} \cup \{ Q\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_\lambda $, $\lambda \in \Lambda $, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementary or $H_\lambda =G$ for some $\lambda \in \Lambda $.

We obtain an upper bound for relative Dehn functions of amalgamated products and HNN--extensions with respect to certain collections of subgroups. Our main results generalize the combination theorems for relatively hyperbolic groups proved by Dahmani.

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{ H_1, \ldots , H_m\} $. We prove that if each of the subgroups $H_1, \ldots , H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$ is also finite.

We show that every finitely generated relatively hyperbolic group is asymptotically tree--graded with respect to the collection of cosets of peripheral subgroups. Together with results of the paper of C. Drutu and M. Sapir, this completes the characterization of relatively hyperbolic groups in terms of their asymptotic cones.

We show that for any metric space $M$ satisfying certain natural conditions, there is a finitely generated group $G$, an ultrafilter $\omega $, and an isometric embedding $\iota $ of $M$ to the asymptotic cone $\C $ such that the induced homomorphism $\iota ^ \ast :\pi_1(M)\to \pi_1(\C )$ is injective. In particular, we prove that any countable group can be embedded into a fundamental group of an asymptotic cone of a finitely generated group.

We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of uniformly exponential growth. We also show that $0$ is an accumulation point of the set of entropies of elementary amenable groups.

In this note we show that many natural properties of ordinary hyperbolic groups can not be generalized to the case of weakly relatively hyperbolic groups. Our main technical tool is a theorem describing weak relatively hyperbolic structure on HNN extensions and amalgamated products.

We show that any finitely generated group of exponential growth has uniform exponential growth.

For a given family of groups $B$, an elementary class $\mathcal E(B)$ is defined as the smallest class of groups, that contains $B$ and is closed under the passing to subgroups, quotients, and taking extensions and direct limits. The properties of these classes are considered. The proposed theory has close connections with the theory of elementary amenable groups as well as of Kuro\v s -- \v Cernikov classes.

Answering a question of Shalom, we provide first examples of non--amenable finitely generated groups $G$ such that the infimum of the Kazhdan constants of $G$ taken over all finite generating sets of $G$ equals $0$.

We show that for every infinite hyperbolic group $H$, the uniform Kazhdan constant of $H$ equals $0$.

Given a finitely presented group $H$, finitely generated subgroup $B$ of $H$, and a monomorphism $\psi :B\to H$, we obtain an upper bound of the Dehn function of the corresponding HNN-extension $G=\langle H, t\; |\; t^{-1}Bt=\psi (B)\rangle $ in terms of the Dehn function of $H$ and the distortion of $B$ in $G$. Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol'shanskii, Rips, and Sapir.

We introduce and study the notion of an exponential radical $Exp(G)$ of a Lie group $G$. In case $G$ is a connected simply--connected solvable Lie group, $Exp (G)$ is a normal Lie subgroup of $G$ and the quotient $G/Exp(G)$ has polynomial volume growth. These results yield that every subgroup of a polyciclic group has either polynomial or expenential relative growth.

An exact formula of the distorrtion of a Lie subgroup in a connected simply--connected nilpotent Lie group is provided. In particular, we show that a function $f(n)=n^r$ can be realized (up to equivalence) as the distortion function of a Lie subgroup into a connected simply--connected nilpotent Lie group if and only if $r\in \mathbb Q$. On the other hand, for homogeneous groups only functions of type $n^r$, where $r\in \mathbb Z$, can be realized. Passing to lattices, we obtain similar results for finitely generated nilpotent Lie groups.

We show that finitely generated (even cyclic) subgroups of finitely generated metabelian linear groups may have intermediate relative growth. We also prove that every function satisfying some natural conditions can be realized as a growth function of a cyclic subgroup in a finitely generated center--by--metabelian group up to a certain equivalence. On the other hand, the relative growth function of every finitely generated subgroup of a finitely generated free solvable group is either polynomial or exponential. In contrast, this result is not longer valid if we consider arbitrary (not necesseraly finitely generated) subgroups.

We introduce and study rank growth functions of subgroups of finitely generated groups. For solvable groups, we obtain estimetas on the growth of rank of subgroups generalizing some previously known results on finitely generated subgroups.

We study the growth of rank of subgroups of finitely generated free group. First we show that for every normal subgroup $N$ of a finitely generated free group $F$, the growth function of rank of $N$ is equivalent to the growth function of $F/N$. On the other hand, any non--decreasing function bounded by some exponential function can be realized as the rank growth function of some subgroup of $F$. We also prove a generalization of the Howson theorem in terms of the growth of ranks of subgroups.