Geometric Group Theory and Topology @ Tufts

The GGTT@Tufts Seminar is held weekly during term time. Seminars are held in the Department of Mathematics, Tufts University.

Unless otherwise noted, the location and time of seminars will be as follows:

Where: Bromfield-Pearson 001, Department of Mathematics, Tufts University (Medford Campus). Directions
When: 4:15pm, Tuesdays in term time.

For more information about GGTT@Tufts, email the Seminar Secretary.

Regular Participants

Regular participants come from universities in the Greater Boston Area. Regular participants include:

Ruth Charney (Brandeis University)
Mauricio Gutierrez (Tufts University)
Boris Hasselblatt (Tufts University)
Ki Hyoung Ko (Korea Advanced Institute of Science and Technology, visitor to Brandeis University)
Max Margolis (Brandeis University)
Zbigniew Nitecki (Tufts University)
Mark O'Brien (Tufts University)
Adam Piggott* (Tufts University)
Andy Putman (MIT)
Kim Ruane (Tufts University)
Michael Shapiro (Tufts University)
Genevieve Walsh (Tufts University)
Richard Weiss (Tufts University)

*=Seminar Secretary

Upcoming Talks

Please note that this schedule is advisory only, and is subject to last minute change. If you plan on travelling to Tufts to hear a particular talk, it is best to contact the Seminar Secretary to confirm details before setting off.

The GGTT seminar has adjourned until Fall 2008.

Past Talks

Edward Taylor (Wesleyan University), "The Quasiconformal Homogeneity of Hyperbolic Manifolds" One can measure the analytic symmetry of a hyperbolic manifold via the study of the collection of quasiconformal automorphisms of the manifold. It turns out that the topology and geometry of quasiconformally homogeneous hyperbolic manifolds is tightly controlled. We will explain the basics of this theory.

Matthew Day (University of Chicago), "Finite generation of subgroups of automorphism groups of right-angled Artin groups" A right-angled Artin group (RAAG) is a group with a finite presentation whose only relations are commutation relations between generators. RAAGs interpolate between free groups and free abelian groups. I will define a class of subgroups of automorphism groups of RAAGs that interpolate between mapping class groups and symplectic groups in a parallel way. I will then sketch a proof that these subgroups are finitely generated. This will include a generalization of the classical peak-reduction theorem of J.H.C. Whitehead.

Keith Burns (Northwestern University)

Ki Hyoung Ko (Korea Advanced Institute of Science and Technology), "Graph braid groups and right-angled Artin groups" We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index greater than or equal to 5.

Tim Riley (University of Bristol), "Hydra Groups" I will describe a new family of groups exhibiting wild geometric and computational features in the context of their Conjugacy Problems. These features stem from manifestations of "Hercules versus the hydra battles." This is joint work with Martin Bridson.

Robert Meyerhoff (Boston College), Topic TBA

Daniel Ruberman (Brandeis University), "Fundamental groups of surface complements"

Genevieve Walsh, "Residually finite rational solvable groups, right-angled Coxeter groups, and the virtually fibered conjecture" I'll discuss a paper of Ian Agol in which he shows that if a 3-manifold has fundamental group which is residually finite rationally solvable (RFRS), then the 3-manifold is virtually fibered. Finitely-generated right-angled Coxeter groups have a finite-index subgroup which is RFRS, and it follows that many 3-manifolds are virtually fibered.

Colin Adams (Williams College), "Quasi-Fuchsian surfaces in noncompact hyperbolic 3-manifolds"

Andrew Putman (MIT), "Subgroup distortion in the mapping class group" I will prove that the Torelli subgroup of the mapping class group is at least exponentially distorted in the mapping class group. This answers a question of Hamenstadt. I will also discuss two kinds of generalizations of this result, one to certain non-normal subgroups which contain the Torelli group and another to any nontrivial finitely generated normal subgroup of the mapping class group which is contained in the Torelli group. This is joint work with N. Broaddus and B. Farb.

James Propp (University of Massachusetts Lowell), "Chip-firing groups and fractal structures arising from graph-Laplacians" The chip-firing game has been invented at least twice: once as a pedagogical tool for teaching probability (the "stochastic abacus") and once as a testbed for ideas about the origins of complexity in the world ("self-organized criticality"). I'll convey the basic facts about chip-firing on directed graphs, focussing on the chip-firing group of a directed graph and its relationship to the Laplacian of the graph, and I'll show how some simple definitions give rise to complicated fractal structures whose nature has so far defied analysis.

Petra Hitzelberger (Universität Münster and Tufts University), "Buildings having octagons as boundary"

Michael Shapiro, "Groups that do not have Cannon's algorithms"

Yvonne Lai (University of California Davis), "An effective compactness theorem for Coxeter groups" Through highly non-constructive methods, works by Bestvina, Culler, Morgan,Paulin, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a small subgroup, then the space of its discrete and faithful actions on H^n, modulo conjugation, is compact for all dimensions. We make this result effective for Coxeter groups. We find that either the group splits over a small subgroup or there is a constant C and a point in H^n that is moved no more than C by any generator.

Ki Hyoung Ko (Korea Advanced Institute of Science and Technology), "A polynomial-time solution to the reducibility problem of braid groups"

Richard Weiss, "Locally finite affine buildings" Roughly speaking, an affine building is a collection of Euclidean spaces of a fixed dimension l amalgamated according to certain rules. Bruhat and Tits classified affine buildings with l ≥ 4 in terms of spherical buildings defined over a field K complete with respect to a discrete valuation ν. One of these buildings is locally finite precisely when the residue field of K with respect to ν is finite. We will give a very brief overview of these results.

Tao Li (Boston College), "Heegaard splittings of amalgamated 3-manifolds and distance in the curve complex"

James W. Cannon (Brigham Young University), "Constructing random 3-manifolds"

Shawn Rafalski (Williams College), "Immersed Hyperbolic Turnovers" Take two copies of a hyperbolic triangle with interior angles &pi/p,&pi/q and &pi/r, where p, q and r are integers, and identify these two triangles together in the natural way along their boundaries. The result is called a hyperbolic turnover, and is a specific example of a two-dimensional hyperbolic orbifold. In this talk, we will see that mapping a turnover by an immersion (which is not an embedding) into a hyperbolic three-orbifold places strong restrictions on the volume of the three-orbifold.

Daniel Ruberman (Brandeis University), "Knot concordance and Heegaard-Floer invariants in branched covers" Heegaard-Floer theory gives rise to an integer-valued invariant tau(K,Y) for a knot K in a 3-manifold Y. Following an approach pioneered by Casson and Gordon in the 1970's, we study the family of invariants tau(K_n,Y_n) where Y_n is the n-fold branched cover of a knot K in the 3-sphere, and K_n is its lift. These invariants give new obstructions to K being slice in the 4-ball.

Kim Ruane, "Automorphisms of Graphs of Groups"

Yoav Segev (Ben Gurion University), "The root groups of special Moufang sets" A Moufang set is a doubly transitive permutation group G on a set X with |X| at least 3, such that the point stabilizer G_x contains a normal subgroup U_x (the root group) which is regular on the remaining points and whose conjugates generate G. Moufang sets should be thought of as rank one Moufang buildings and as such they are the basic building blocks of Moufang buildings. Special Moufang sets (or more precisely special abstract rank one groups) were given a considerable amount of attention in Timmesfelds book ("Abstract Root Subgroups and Simple Groups of Lie-Type"). I will discuss, and give partial results on the conjecture that says that a special Moufang set has abelian root groups.

Patrick Bahls (UNC Asheville), "Asymptotic connectivity and the geometry of a graph"

Dmitriy Sonkin (University of Virginia), "Limits of hyperbolic groups and uniform Kazhdan groups" I will discuss geometric method of graded van Kampen diagrams and, as an application, a construction of infinite groups that admit positive uniform Kzhdan constant.

Stephan Tillmann (University of Melbourne), "The Thurston norm via normal surfaces" I will describe an algorithm to compute the unit ball of the Thurston norm using normal surface theory. Applications include an algorithm to decide whether a 3-manifold fibres over the circle. This is joint work with Daryl Cooper.

Sarah Rees (University of Newcastle-Upon-Tyne), "Group geodesics: regularity, star-freedom and local testability", 3:00pm The geodesics words in a finitely generated group are known to form a regular set whenever the group is either word hyperbolic or free abelian. For selected generating sets, the same is true for virtually abelian groups, geometrically finite hyperbolic groups, all Coxeter groups, Artin groups of finite type and indeed all Garside groups; this list does not claim to be exhaustive. I report on an investigation to look for connections between algebraic properties of such a group, combinatorial properties of its presentations, the structure of its regular set of geodesics, and the complexity of its word problem. That work is joint work with Gilman, Hermiller and Holt. Terms such as regularity, star-freedom and local testability will be defined in the talk; each can be shown to have several different disguises (set-theoretic, geometric, or algebraic, in terms of an associated finite semigroup). We shall see in particular that certain small cancellation conditions on a presentation (which imply word hyperbolicity) force the set of geodesic words to be star-free, that a rather restrictive (but also natural) form of local testability of geodesics implies that the word problem for that group is context-free, and hence characterises virtually free groups, that 1-local testability characterises free abelian groups, and that in general a group with locally testable geodesics can have only finitely many conjugacy classes of torsion elements.
Jason Behrstock (University of Utah), "Dimension and rank of mapping class groups", 4:30pm We will discuss work with Yair Minsky towards understanding the large scale geometry of the mapping class group. In particular, we'll explain how to obtain various topological properties of the asymptotic cone of the mapping class group including a computation of its dimension. An application of this analysis is an affirmative solution to Brock-Farb's Rank Conjecture which asserts that MCG has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup.

Adam Piggott, "Rigidity of graph products of abelian groups" We present a report on joint work with Mauricio Gutierrez. We show that if a group G has a graph-product decomposition with finitely-generated abelian vertex groups, then there are two canonical graph-product decompositions of G.

Jennifer Taback (Bowdoin), "Asymptotic density of subgroups of Thompson's group F" I will discuss the notion of "asymptotic subgroups" of a finitely generated group G. Consider the set of all k-generated subgroups of G, and stratify this set in some way. Count the number of subgroups isomorphic to a given subgroup in each layer of the stratification. This yields a ratio, whose limit is the density of that subgroup. If the limit is 1, we say the subgroup is "generic" and if it is positive, we say the subgroup is "asymptotically visible". One motivation for these definitions arises in cryptography. Thompson's group F has recently been proposed as a platform for an algebraic cryptosystem. While this initial attempt was not secure, it is interesting to consider whether this group has any potential in this way. As a result, one is interested in the densities of k-generated subgroups of this group. We prove that any k-generated subgroup of F is visible with respect to a particular stratification. I will discuss the proof of this theorem as well as the advantages and disadvantages of the particular stratification we use.

Matthew Hedden (MIT), "Lens space surgeries and complex curves" In this talk I'll prove that if positive Dehn surgery on a knot in the three-sphere is a lens space, then that knot arises as the transverse intersection of a complex curve in the four-ball with the three-dimensional sphere. In this way, knots admitting lens space surgeries are similar to links of singularities ( i.e. positive torus knots and sufficiently positive iterated torus knots). However, the complex curves in the case of lens space surgeries may have more interesting singular loci. The proof of this fact uses an invariant of smooth knot concordance defined in the context of Ozsvath-Szabo Floer homology. The above will be a corollary of the main theorem which shows that this concordance invariant detects when fibered knots arise from complex curves with a certain genus constraint. To prove these theorems, I'll make use of connections between Ozsvath-Szabo theory and a branch of geometry called contact geometry (due to Peter Ozsvath and Zoltan Szabo), between contact geometry and the topology of open book decompositions (due to Emannuel Giroux and others), and the relationship between the knot theory of complex curves and the braid group (due to Lee Rudolph, and Michel Boileau and Stepan Orevkov). The theorem about knots admitting lens space surgeries will also make use of work of Paolo Ghiggini and Yi Ni which show that the Ozsvath-Szabo knot invariants detect fibered knots in the three-sphere.

Petra Hitzelberger (Universität Münster), "A convexity theorem for affine buildings" Many versions of Kostant's convexity theorem exist, which originally was proven for semisimple Lie-groups. We show a similar theorem for affine buildings involving two different retractions onto a fixed apartment. The proof can be reduced to a problem in coxeter complexes which we solve using a character fomula for highest weight representations of algebraic groups.

Genevieve Walsh, "Commensurability and Knot Complements"

Stefan Friedl (Université du Québec à Montréal), "Symplectic 4-manifolds and subgroup separability" In 1976 Thurston showed that if N is a fibered 3-manifold, then the 4-manifold S^1 x N is symplectic. We will show that the converse holds if N satisfies certain subgroup separability properties.

Martin Bridgeman (Boston College), "Hausdorff dimension under bending deformations and the Weil-Petersson metric" We analyse how the Hausdorff dimension of the limit set of a Kleinian group changes near the fuchsian locus in quasifuchsian space of a surface. We describe a new metric on Teichmüller space obtained by taking the second derivative of Hausdorff dimension and show that this new metric is bounded below by the classical Weil-Petersson metric. We use this to relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil-Petersson metric of the associated earthquake vector of the lamination. This is joint work with Ed Taylor.

Mark O'Brien, "STUF" As alluded to in the previous lecture, there are many differences between an arbitrary right-angled Coxeter pair and the Davis complex of the corresponding right-angled Coxeter system. The biggest difference is the number of connected components the complement of a fixed point set can have. Other differences include convexity of components, the relationship between the fixed point sets of the simple reflections and their products, and how the fixed points sets of simple reflections sit in the space relative to each other. In this talk I will discuss how in some sense these are the only differences involved, and they arise only in the presence of certain kinds of automorphisms of the acting right-angled Coxeter group.

Mark O'Brien, "Right-angled Coxeter Pairs" A strict fundamental domain allows one to understand a space as a whole from a compact region and stabilizer information. I give rather mild conditions which guarantee that the action of a right-angled Coxeter group has a strict fundamental domain. Applications include comparisons to the Davis complex, automorphisms of right-angled Coxeter groups, and boundary information.

Adam Piggott, "Presenting the Automorphism Group of a 'Tree Product' of Primary Cyclic Groups." We show how to write down a finite presentation for Aut W in case W is a graph product of primary cyclic groups determined by a tree. This seminar continues the story of the previous seminar, but will be independent---so come along even if you missed last weeks show.

Adam Piggott, "On the Automorphisms of Graph Products of Cyclic Groups." We present a report on joint work with Mauricio Gutierrez. Graph products of cyclic groups include free groups, finitely generated abelian groups, right-angled Artin groups and right-angled Coxeter Groups. A unified treatment of the automorphism groups of such groups appears doomed, but we are able to learn much by restricting our attention to a natural subgroup of the automorphism group.

Mauricio Gutierrez, "The conjugating automorphisms of a graph products of groups" We will present a result of Michael Laurence that the group of conjugating automorphisms of a graph product of groups is generated by the set of locally-inner automorphisms.

Ruth Charney, "Relative hyperbolicity and Artin groups" If a group acts geometrically on a hyperbolic metric space, then it is word hyperbolic. We introduce a notion of a "weakly geometric" action and show that a group acting weakly geometrically on a hyperbolic metric space is (weakly) relatively hyperbolic with respect to a finite set of isotropy subgroups. In particular, we apply this to Artin groups where we conjecture that an Artin group is weakly hyperbolic relative to its spherical parabolics if and only if the associated Coxeter group is word hyperbolic. This is joint work with John Crisp.

Michael Shapiro, "Dehn's algorithms for non-hyperbolic groups II" See abstract below.

Michael Shapiro, "Dehn's algorithms for non-hyperbolic groups." Given a finitely generated group G, a Dehn's algorithm is a finite set of length reducing rules u -> v with the property that these reduce a word to the empty word if and only if that word represents the identity. If we take the words u and v to be written in the generators of G, the G has a Dehn's algorithm if and only if it is word hyperbolic. By allowing letters outside the group, we can produce Dehn's algorithms for a much larger class of groups.

Kim Ruane, "Relative Hyperbolicty II." Part II of an introduction to Relatively Hyperbolic Groups.

Murray Elder (Stevens Institute of Technology), "D-finiteness and Group Growth." In combinatorics, people like to count things, which gives rise to sequences of integers. Given a sequence of integers c_0,c_1,c_2,.... people like to consider the formal power series g(z)= c_0+c_1*z+c_2*z^2+... where z is just some formal (complex) variable. With some luck, this function g(z) can turn out to be nice, it is sometimes rational, sometimes algebraic, and failing that, it could be "d-finite". We'll define d-finite in the talk. In the group theory world, what we like to count most is the number of elements in the ball/sphere of radius n in the Cayley graph of a group with fixed generating set. This number is called the "growth function", and the formal powers series you get from it is called the "growth series" for the group and generating set. Lots of work has been done on describing the actual growth functions - they can be polynomial, exponential, or "intermediate" (Grigorchuk proved there were groups with growth faster than polynomial but sub-exponential). But not so much on the growth series, unless it was rational. In this talk, we consider the different types of growth series that can arise for different groups - rational, algebraic, d-finite and not d-finite. The talk should be accessible to a general math audience.

Kim Ruane, "Relative Hyperbolicty." Part I of an introduction to Relatively Hyperbolic Groups.

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