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At least 19 recordsLinked to original sources
arXiv2007

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups, in two different contexts. First, we elaborate on a remark from math.GR/0601311, which states that the version of Dehn filling in relatively hyperbolic groups proved in math.GR/0510195, allowing infinitely generated parabolic subgroups, follows from the version with finitely generated parabolics. Second, we observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We use this (and known results) to derive some consequences about the Strong Novikov Conjecture for groups as constructed in math.GR/0411039.

math.GR↗
arXiv2008

Erratum on "Hadamard spaces with isolated flats"

The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats are extended to a more general setting in which the isolated subspaces are not necessarily flats. However, one step of that extension does not follow from the argument we used the isolated flats setting. We provide a new proof that fills this gap. In addition, we give a more detailed account of several other parts of Theorem A.0.1, which were sketched in math.GR/0411232.

math.GR↗
arXiv2012

A note on ampleness in the theory of non abelian free groups

Recently Ould Houcine-Tent (see arXiv:1205.0929v2 [math.GR]) proved that the theory of non abelian free groups is $n$-ample for any $n<ω$. We give a sequence of single elements in $F_ω$ witnessing the above mentioned result. Our proof is not independent from the one given in arXiv:1205.0929v2 [math.GR], as we essentially use some theorems from there. On the other hand our witnessing sequence is much simpler.

math.LO↗
arXiv2004

Diagrams and the second homotopy group

We use Klyachko's methods [A funny property of sphere and equations over groups, Comm. in Alg. 21 (1993) 2555--2575] (see also Fenn-Rourke, L'Enseignment Math. 42 (1996) 49--74 and math.GR/9810184 and Cohen-Rourke, math.GR/0009101) to prove that, if a 1-cell and a 2-cell are added to a complex with torsion-free fundamental group, and with the 2-cell attached by an amenable t-shape, then pi_2 changes by extension of scalars. It then follows using a result of Bogley and Pride, Proc. Edinburgh Math. Soc. 35 (1992) 1--39, that the resulting fundamental group is also torsion free. We also prove that the normal closure of the attaching word contains no words of smaller complexity.

math.AT↗
arXiv2005

Rigidity of hyperbolic P-manifolds: a survey

In this survey paper, we outline the proofs of the rigidity results for simple, thick, hyperbolic P-manifolds found in our three earlier papers math.GR/0506518, math.GT/0410476, and math.GR/0409586. We discuss how the arguments change in the two, three, and higher dimensional settings. This paper was written for the 22nd Annual Workshop in Geometric Topology (Colorado Springs, June 2005).

math.GT↗
arXiv2008

The well-ordering of dual braid monoids

We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:0811.3902 [math.GR], and then proving that this new ordering coincides with the standard ordering of braids.

math.GR↗
arXiv2012

An introduction to hyperlinear and sofic groups

This is an edited write-up of lecture notes of the 7-th Appalachian set theory workshop of the same title led by the first named author at the Cornell University on November 22, 2008. A draft version of the notes was prepared by the second named author. This presentation is largely complementary to the earlier survey by the first-named author (Hyperlinear and sofic groups: a brief guide, Bull. Symb. Logic 14 (2008), pp. 449-480; arXiv:0804.3968v8 [math.GR]).

math.GR↗
arXiv2000

Dimension $n$ Representations of the Braid Group on $n$ Strings

In 1996 E. Formanek classified all the irreducible complex representations of $B_n$ of dimension at most $n-1,$ where $B_n$ is the Artin braid group on $n$ strings. In this paper we extend this classification to the representations of dimension $n,$ for $n\geq 9$. We prove that all such representations are equivalent to a tensor product of a one-dimensional representation and a specialization of a certain one-parameter family of $n-$dimensional representations, that was first discovered in 1996 by Tong, Yang, and Ma. We use our classification of irreducible complex representations of corank two, in the preprint math.GR/0003047.

math.GR↗
arXiv2000

Philip Hall's Problem On Non-Abelian Splitters

Philip Hall raised around 1965 the following question which is stated in the Kourovka Notebook: Is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will decompose the problem into two parts: We want to find non-commutative splitters, that are groups G not= 1 with Ext(G,G)=1 . The class of splitters fortunately is quite large so that extra properties can be added to G. We can consider groups G with the following properties: There is a complete group L with cartesian product L^omega cong G, Hom(L^omega,S_omega)=0 (S_omega the infinite symmetric group acting on omega) and End(L,L)=Inn(L) cup {0}. We will show that these properties ensure that G is a splitter and hence obviously a Hall-group in the above sense. Then we will apply a recent result from our joint paper math.GR/0009089 which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.

math.GR↗
arXiv2006

Reflection group of the quaternionic Lorentzian Leech lattice

In this article we study a second example of the phenomenon studied in "Complex Lorentzian Leech lattice and bimonster".(Arxiv. math.GR/0508228). The results and methods of proof are similar. We find 14 roots in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram D given by the incidence graph of projective plane over finite field of order 2. We prove that the reflections in these 14 roots generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the reflection group. Much of the work follows the analogy that D is like the Coxeter-Dynkin diagram for this hyperbolic reflection group.

math.GR↗
arXiv2005

F-quasigroups and generalized modules

In math.GR/0510298, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the equational class of (pointed) F-quasigroups and the equational class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.

math.GR↗
arXiv2006

F-quasigroups isotopic to groups

In math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally, we show an equivalence of equational classes between pointed FG-quasigroups and central generalized modules over a particular ring.

math.GR↗
arXiv2008

On amenability of group algebras, II: graded algebras

This paper continues math.GR/0608302's study of amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and applies it to graded algebras associated with finitely generated groups. Due to a mistake in Lemma 4.3, the main statements of the present paper should not be considered as proven. The results in Sections 2 and 3 are nevertheless valid.

math.GR↗
arXiv2006

Ore revisited: an algorithmic investigation of the simple commutator promise problem

Motivated by a desire to test the security of the pubic key exchange protocol of I. Anshel, M. Anshel, and D. Goldfeld, (``An Algebraic Method for Public-Key Cryptography'', Mathematical Research Letters, vol. 6, pp. 1-5, 1999), we study algorithmic approaches to the simple commutator decision and promise problems (SCDP/SCPP) for the braid groups B_n. We take as our point of departure a seminal paper of O. Ore, (``Some Remarks on Commutators'', Proceedings of the American Mathematical Society, Vol. 2, No. 2, pp.307-314, 1951), which studies the SCPP for the symmetric groups. Our results build on the work of H. Cejtin and I. Rivin, (``A Property of Alternating Groups'', arXiv:math.GR/0303036).

math.GR↗
arXiv2001

On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finite-index subgroups, they also hold for the profinite completions $\hat G$ of the groups G. The representations involved have interesting spectral properties investigated in math.GR/9910102. This paper serves as a group-theoretic counterpart to the studies in the mentionned paper. We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsion-free branch just-infinite group. We also produce a new example of branch just-infinite group of intermediate growth, and provide for it an L-type presentation by generators and relators.

math.GR↗