10 research outputs found
Groups of given intermediate word growth
We show that there exists a finitely generated group of growth ~f for all
functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq
f(\eta R) for all R large enough and \eta\approx2.4675 the positive root of
X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than
\exp(R^{\log2/\log\eta}).
We also give a family of self-similar branched groups of growth
~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).Comment: small typos corrected from v
Growth behaviors in the range
For every in a left neighborhood of 1, a
group is constructed, the growth function of which satisfies
and . When ,
this provides an explicit uncountable collection of groups with growth
functions strictly comparable. On the other hand, oscillation in the case
explains the existence of groups with non comparable growth
functions. Some period exponents associated to the frequency of oscillation
provide new group invariants.Comment: Final version to appear in Afrika Matematik
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