arXiv subjects

Sean Cleary

Publications and source records attributed to Sean Cleary.

35 records · Page 2Linked to original sources

arXiv2009

Random subgroups of Thompson's group $F$

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results.

math.GR↗

arXiv2008

Metric properties of higher-dimensional Thompson's groups

Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.

math.GR↗

arXiv2007

Metric properties of the braided Thompson's groups

Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.

math.GR↗

arXiv2007

A finitely presented group with unbounded dead-end depth

The dead-end depth of an element g of a group G, with respect to a generating set A is the distance from g to the complement of the radius $d_A(1,g)$ closed ball, in the word metric $d_A$ defined with respect to A. We exhibit a finitely presented group G with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements. The authors regret that the published version of this article (Proc. Amer. Math. Soc., 134(2), pp.343-349, 2006) contains a significant error concerning the model for G described in Section 2. We are grateful to Jorg Lehnert for pointing out our mistake. In this corrected version, that model has been overhauled, and that has necessitated a number of changes in the subsequent arguments.

math.GR↗

arXiv2006

The word problem distinguishes counter languages

Counter automata are more powerful versions of finite-state automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of $\Z^n$ is accepted by a nondeterministic $m$-counter automaton if and only if $m \geq n$.

math.GR↗

arXiv2006

Bounding right-arm rotation distances

Rotation distance measures the difference in shape between binary trees of the same size by counting the minimum number of rotations needed to transform one tree to the other. We describe several types of rotation distance where restrictions are put on the locations where rotations are permitted, and provide upper bounds on distances between trees with a fixed number of nodes with respect to several families of these restrictions. These bounds are sharp in a certain asymptotic sense and are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

math.GR↗

arXiv2006

Pure braid subgroups of braided Thompson's groups

We describe pure braided versions of Thompson's group F. These groups, $BF$ and $\hat{BF}$, are subgroups of the braided versions of Thompson's group V, introduced by Brin and Dehornoy. Unlike V, elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe infinite and finite presentations of these groups.

math.GR↗

arXiv2005

Computational explorations in Thompson's group F

We describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson's group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

math.GR↗

arXiv2005

Distortion of wreath products in some finitely presented groups

Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian group G. We find that Z wr Z is undistorted in F but is at least exponentially distorted in G.

math.GR↗

arXiv2005

Cone types and geodesic languages for lamplighter groups and Thompson's group F

We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups $L_n$ have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and no regular language of geodesics with respect to the standard finite generating set. We show that the existence of families of "seesaw" elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.

math.GR↗

arXiv2004

Metric properties of the lamplighter group as an automata group

We examine the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group due to Grigorchuk and Zuk. We find some metric behavior with respect to this generating set analogous to the metric behavior in the standard wreath product generating set. The similar metric behavior includes expressions for geodesic paths and families of `dead-end' elements, which are endpoints of terminating geodesic rays. We also exhibit some different metric behavior between these two generating sets related to the existence of `seesaw' elements.

math.GR↗

arXiv2004

Seesaw words in Thompson's group F

We describe a family of words in Thompson's group F which present a challenge to the question of finding canonical minimal length representatives, and which show that F is not combable by geodesics. These words have the property that there are only two possible suffixes of long lengths for geodesic paths to the word from the identity; one is of the form $g^k$ and the other of the form $g^{-k}$ where g is a generator of the group.

math.GR↗

arXiv2003

Dead end words in lamplighter groups and other wreath products

We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element $w$ in a group $G$ with finite generating set $X$ is a dead end element if no geodesic ray from the identity to $w$ in the Cayley graph $Γ(G,X)$ can be extended past $w$. Additionally, we describe some nonconvex behavior of paths between elements in these Cayley graphs and seesaw words, which are potential obstructions to these graphs satisfying the $k$-fellow traveller property.

math.GR↗

arXiv2002

Geometric quasi-isometric embeddings into Thompson's group F

We use geometric techniques to investigate several examples of quasi-isometrically embedded subgroups of Thompson's group F. Many of these are explored using the metric properties of the shift map phi in F. These subgroups have simple geometric but complicated algebraic descriptions. We present them to illustrate the intricate geometry of Thompson's group F as well as the interplay between its standard finite and infinite presentations. These subgroups include those of the form F^m cross Z^n, for integral non-negative m and n, which were shown to occur as quasi-isometrically embedded subgroups by Burillo and Guba and Sapir.

math.GR↗

arXiv2002

Combinatorial properties of Thompson's group F

We study some combinatorial properties of the word metric of Thompson's group F in the standard two generator finite presentation. We explore connections between the tree pair diagram representing an element w of F, its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of F with respect to the two generator finite presentation. Namely, we exhibit the form of `dead end' elements in this Cayley graph, and show that it has no `deep pockets'. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.

math.GR↗

arXiv2002

Thompson's group F is not almost convex

We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any integer n in the standard finite two generator presentation. To accomplish this, we construct a family of pairs of elements at distance n from the identity and distance 2 from each other, which are not connected by a path lying inside the n-ball of length less than k for increasingly large k. Our techniques rely upon Fordham's method for calculating the length of a word in F and upon an analysis of the generators' geometric actions on the tree pair diagrams representing elements of F.

math.GR↗

arXiv1998

Metrics and embeddings of generalizations of Thompson's group F

The distance from the origin in the word metric for generalizations F(p) of Thompson's group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F(p). This interpretation of the metric is used to prove that every F(p) admits a quasi-isometric embedding into every F(q), and also to study the behavior of the shift maps under these embeddings.

math.GR↗