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Search Results: 1 - 10 of 7360 matches for " Adam Piggott "
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Palindromic primitives and palindromic bases in the free group of rank two
Adam Piggott
Mathematics , 2005,
Abstract: The present paper records more details of the relationship between primitive elements and palindromes in F_2, the free group of rank two. We characterise the conjugacy classes of primitive elements which contain palindromes as those which contain cyclically reduced words of odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F_2. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.
Detecting the growth of free group automorphisms by their action on the homology of subgroups of finite index
Adam Piggott
Mathematics , 2004,
Abstract: We prove that if F is a finitely generated free group and f:F -> F is an automorphism with polynomial growth of degree d, then there exists a characteristic subgroup S < F of finite index such that the induced automorphism of the abelianisation of S also grows polynomially of degree d. The proof is geometric in nature and makes use of improved relative train track representatives.
Algorithmic constructions and primitive elements in the free group of rank 2
Adam Piggott
Mathematics , 2005,
Abstract: The centrepiece of this paper is a normal form for primitive elements which facilitates the use of induction arguments to prove properties of primitive elements. The normal form arises from an elementary algorithm for constructing a primitive element p in F(x, y) with a given exponent sum pair (X, Y), if such an element p exists. Several results concerning the primitive elements of F(x, y) are recast as applications of the algorithm and the normal form.
Rigidity of graph products of abelian groups
Mauricio Gutierrez,Adam Piggott
Mathematics , 2007,
Abstract: We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
The Bieri-Neumann-Strebel Invariant of the Pure Symmetric Automorphisms of a Right-Angled Artin Group
Nic Koban,Adam Piggott
Mathematics , 2013,
Abstract: We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.
On the automorphisms of a graph product of abelian groups
Mauricio Gutierrez,Adam Piggott,Kim Ruane
Mathematics , 2007,
Abstract: We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.
The automorphism group of the free group of rank two is a CAT(0) group
Adam Piggott,Kim Ruane,Genevieve S. Walsh
Mathematics , 2008,
Abstract: We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
Recognizing Right-Angled Coxeter Groups Using Involutions
Charles Cunningham,Andy Eisenberg,Adam Piggott,Kim Ruane
Mathematics , 2014,
Abstract: We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a result stating that if the defining graph contains no SILs, then Aut^0(W) is a right-angled Coxeter group.
$\mathrm{CAT}(0)$ Extensions of Right-angled Coxeter Groups
Charles Cunningham,Andy Eisenberg,Adam Piggott,Kim Ruane
Mathematics , 2015,
Abstract: We show that any split extension of a right-angled Coxeter group $W_{\Gamma}$ by a generating automorphism of finite order acts faithfully and geometrically on a $\mathrm{CAT}(0)$ metric space.
9-Hydroxyfurodysinin-O-ethyl Lactone: A New Sesquiterpene Isolated from the Tropical Marine Sponge Dysidea arenaria
A. Piggott,P. Karuso
Molecules , 2005, DOI: 10.3390/10101292
Abstract: A new sesquiterpene, 9-hydroxyfurodysinin-O-ethyl lactone, has been isolated from a New Caledonian Dysidea arenaria, along with three known compounds. The possible incorporation of the ethyl ether from the extraction solvent is discussed.
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