783 research outputs found
Dimension and randomness in groups acting on rooted trees
We explore the structure of the p-adic automorphism group Gamma of the
infinite rooted regular tree. We determine the asymptotic order of a typical
element, answering an old question of Turan.
We initiate the study of a general dimension theory of groups acting on
rooted trees. We describe the relationship between dimension and other
properties of groups such as solvability, existence of dense free subgroups and
the normal subgroup structure. We show that subgroups of Gamma generated by
three random elements are full-dimensional and that there exist finitely
generated subgroups of arbitrary dimension. Specifically, our results solve an
open problem of Shalev and answer a question of Sidki
The Hanna Neumann Conjecture and the rank of the join
The Hanna Neumann conjecture gives a bound on the intersection of finitely
generated subgroups of free groups. We explore a natural extension of this
result, which turns out to be true only in the finite index case, and provide
counterexamples for the general case. We also see that the graph-based method
of generating random subgroups of free groups developed by Bassino, Nicaud and
Weil is well-suited to generating subgroups with non-trivial intersections. The
same method is then used to generate a counterexample to a similar conjecture
of Guzman.Comment: 9 pages, 4 figures; added counterexample to a conjecture of Guzma
Invariable Generation of Infinite Groups
A subset S of a group G invariably generates G if G = for
each choice of g(s) in G, s in S. In this paper we study invariable generation
of infinite groups, with emphasis on linear groups. Our main result shows that
a finitely generated linear group is invariably generated by some finite set of
elements if and only if it is virtually solvable. We also show that the
profinite completion of an arithmetic group having the congruence subgroup
property is invariably generated by a finite set of elements
Aspects of Nonabelian Group Based Cryptography: A Survey and Open Problems
Most common public key cryptosystems and public key exchange protocols
presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve
methods are number theory based and hence depend on the structure of abelian
groups. The strength of computing machinery has made these techniques
theoretically susceptible to attack and hence recently there has been an active
line of research to develop cryptosystems and key exchange protocols using
noncommutative cryptographic platforms. This line of investigation has been
given the broad title of noncommutative algebraic cryptography. This was
initiated by two public key protocols that used the braid groups, one by Ko,
Lee et.al.and one by Anshel, Anshel and Goldfeld. The study of these protocols
and the group theory surrounding them has had a large effect on research in
infinite group theory. In this paper we survey these noncommutative group based
methods and discuss several ideas in abstract infinite group theory that have
arisen from them. We then present a set of open problems
On the genericity of Whitehead minimality
We show that a finitely generated subgroup of a free group, chosen uniformly
at random, is strictly Whitehead minimal with overwhelming probability.
Whitehead minimality is one of the key elements of the solution of the orbit
problem in free groups. The proofs strongly rely on combinatorial tools,
notably those of analytic combinatorics. The result we prove actually depends
implicitly on the choice of a distribution on finitely generated subgroups, and
we establish it for the two distributions which appear in the literature on
random subgroups
Generic complexity of the Conjugacy Problem in HNN-extensions and algorithmic stratification of Miller's groups
We discuss time complexity of The Conjugacy Problem in HNN-extensions of
groups, in particular, in Miller's groups. We show that for "almost all", in
some explicit sense, elements, the Conjugacy Problem is decidable in cubic
time. It is worth noting that the Conjugacy Problem in a Miller group may have
be undecidable. Our results show that "hard" instances of the problem comprise
a negligibly small part of the group
Stable group theory and approximate subgroups
We note a parallel between some ideas of stable model theory and certain
topics in finite combinatorics related to the sum-product phenomenon. For a
simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X|
bounded is close to a finite subgroup, or else to a subset of a proper
algebraic subgroup of G. We also find a connection with Lie groups, and use it
to obtain some consequences suggestive of topological nilpotence. Combining
these methods with Gromov's proof, we show that a finitely generated group with
an approximate subgroup containing any given finite set must be
nilpotent-by-finite. Model-theoretically we prove the independence theorem and
the stabilizer theorem in a general first-order setting.Comment: Further local corrections, thanks to two anonymous referre
Non-unitarisable representations and random forests
We establish a connection between Dixmier's unitarisability problem and the
expected degree of random forests on a group. As a consequence, a residually
finite group is non-unitarisable if its first L2-Betti number is non-zero or if
it is finitely generated with non-trivial cost. Our criterion also applies to
torsion groups constructed by D. Osin, thus providing the first examples of
non-unitarisable groups not containing a non-Abelian free subgroup
Uniform rank gradient, cost and local-global convergence
We analyze the rank gradient of finitely generated groups with respect to
sequences of subgroups of finite index that do not necessarily form a chain, by
connecting it to the cost of p.m.p. actions. We generalize several results that
were only known for chains before. The connection is made by the notion of
local-global convergence.
In particular, we show that for a finitely generated group with
fixed price , every Farber sequence has rank gradient . By adapting
Lackenby's trichotomy theorem to this setting, we also show that in a finitely
presented amenable group, every sequence of subgroups with index tending to
infinity has vanishing rank gradient.Comment: Corrected typing mistakes, added comments. 25 page
Soluble groups with no sections
In this article, we examine how the structure of soluble groups of infinite
torsion-free rank with no section isomorphic to the wreath product of two
infinite cyclic groups can be analysed. As a corollary, we obtain that if a
finitely generated soluble group has a defined Krull dimension and has no
sections isomorphic to the wreath product of two infinite cyclic groups then it
is a group of finite torsion-free rank. There are further corollaries including
applications to return probabilities for random walks. The paper concludes with
constructions of examples that can be compared with recent constructions of
Brieussel and Zheng
- …