27 research outputs found
Algorithmic decidability of Engel's property for automaton groups
We consider decidability problems associated with Engel's identity
( for a long enough commutator sequence) in groups
generated by an automaton. We give a partial algorithm that decides, given
, whether an Engel identity is satisfied. It succeeds, importantly, in
proving that Grigorchuk's -group is not Engel. We consider next the problem
of recognizing Engel elements, namely elements such that the map
attracts to . Although this problem seems intractable in
general, we prove that it is decidable for Grigorchuk's group: Engel elements
are precisely those of order at most . Our computations were implemented
using the package FR within the computer algebra system GAP
Bundling by volume exclusion in non-equilibrium spaghetti
In physical networks, like the brain or metamaterials, we often observe local
bundles, corresponding to locally aligned link configurations. Here we
introduce a minimal model for bundle formation, modeling physical networks as
non-equilibrium packings of hard-core 3D elongated links. We show that growth
is logarithmic in time, in stark contrast with the algebraic behavior of lower
dimensional random packing models. Equally important, we find that this slow
kinetics is metastable, allowing us to analytically predict an algebraic growth
due to the spontaneous formation of bundles. Our results offer a mechanism for
bundle formation resulting purely from volume exclusion, and provide a
benchmark for bundling activation and growth during the assembly of physical
networks
Primitive Words, Free Factors and Measure Preservation
Let F_k be the free group on k generators. A word w \in F_k is called
primitive if it belongs to some basis of F_k. We investigate two criteria for
primitivity, and consider more generally, subgroups of F_k which are free
factors.
The first criterion is graph-theoretic and uses Stallings core graphs: given
subgroups of finite rank H \le J \le F_k we present a simple procedure to
determine whether H is a free factor of J. This yields, in particular, a
procedure to determine whether a given element in F_k is primitive.
Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from
the direct product of k copies of G to G), where G is an arbitrary finite
group. We call w measure preserving if given uniform measure on G x G x ... x
G, w induces uniform measure on G (for every finite G). This is the second
criterion we investigate: it is not hard to see that primitivity implies
measure preservation and it was conjectured that the two properties are
equivalent. Our combinatorial approach to primitivity allows us to make
progress on this problem and in particular prove the conjecture for k=2.
It was asked whether the primitive elements of F_k form a closed set in the
profinite topology of free groups. Our results provide a positive answer for
F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I:
A New Algorithm", and "On Primitive Words II: Measure Preservation". 42
pages, 14 figures. Some parts of the paper reorganized towards publication in
the Israel J. of Mat
Branch Rings, Thinned Rings, Tree Enveloping Rings
We develop the theory of ``branch algebras'', which are infinite-dimensional
associative algebras that are isomorphic, up to taking subrings of finite
codimension, to a matrix ring over themselves. The main examples come from
groups acting on trees.
In particular, for every field k we construct a k-algebra K which (1) is
finitely generated and infinite-dimensional, but has only finite-dimensional
quotients;
(2) has a subalgebra of finite codimension, isomorphic to ;
(3) is prime;
(4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2;
(5) is recursively presented;
(6) satisfies no identity;
(7) contains a transcendental, invertible element;
(8) is semiprimitive if k has characteristic ;
(9) is graded if k has characteristic 2;
(10) is primitive if k is a non-algebraic extension of GF(2);
(11) is graded nil and Jacobson radical if k is an algebraic extension of
GF(2).Comment: 35 pages; small changes wrt previous versio
C*-simplicity and the unique trace property for discrete groups
In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to settle the longstanding open problem of characterizing groups with the unique trace property
Congruence subgroup growth of arithmetic groups in positive characteristic
We prove a new uniform bound for subgroup growth of a Chevalley group G over the local ring double-struck F sign [[t]] and also over local pro-p rings of higher Krull dimension. This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of SLn(Fp[t]) (n ≥ 3) is of type nlog n. This was one of the main problems left open by A. Lubotzky in his article. The essential tool for proving the results is the use of graded Lie algebras. We sharpen Lubotzky's bounds on subgroup growth via a result on subspaces of a Chevalley Lie algebra L over a finite field double-struck F sign. This theorem is proved by algebraic geometry and can be modified to obtain a lower bound on the codimension of proper Lie subalgebras of L