319 research outputs found
Verbal subgroups of hyperbolic groups have infinite width
Let be a non-elementary hyperbolic group. Let be a group word such
that the set of all its values in does not coincide with or 1.
We show that the width of verbal subgroup is infinite. That is,
there is no such that any can be represented as a
product of values of and their inverses.Comment: To appear in Journal of the London Mathematical Society. 22 pages, 8
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Algebraic properties of profinite groups
Recently there has been a lot of research and progress in profinite groups.
We survey some of the new results and discuss open problems. A central theme is
decompositions of finite groups into bounded products of subsets of various
kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
Word maps in Kac-Moody setting
The paper is a short survey of recent developments in the area of word maps
evaluated on groups and algebras. It is aimed to pose questions relevant to
Kac--Moody theory.Comment: 16 pag
Stable W-length
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. In particular, we show that for suitable W, these
asymptotic cones contain certain subgroups that are normed vector spaces.Comment: 24 pages; version 2 incorporates referee's comment
Geometry of word equations in simple algebraic groups over special fields
This paper contains a survey of recent developments in investigation of word
equations in simple matrix groups and polynomial equations in simple
(associative and Lie) matrix algebras along with some new results on the image
of word maps on algebraic groups defined over special fields: complex, real,
p-adic (or close to such), or finite.Comment: 44 page
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