Buffalo Geometry and Topology Seminar

Fall 2008

Unless noted, all seminars are Friday at 4pm, in Mathematics 122.

You may also be interested in the graduate student topology seminar.

Date Speaker Organization Title Abstract

September 5th Organizational Meeting

September 12th Chris Leininger U.I.U.C. Flat structures via length functions I'll discuss joint work with M. Duchin and K. Rafi in which we study flat structures on a closed surface S using lengths of closed geodesics on S.

September 19th Bernard Badzioch University at Buffalo Localizations of mapping spaces. Homotopy localizations are functors which can be interpreted as projections of the category of topological spaces onto its subcategory. This class of functors includes Postnikov sections, homological localizations, Quillen plus construction etc.
One of the central questions one encounters while dealing with such functors is which topological properties of spaces they preserve. The talk will discuss this problem in relation to preservation of the structure of mapping spaces Map(A, X) for a given space A.
This is a joint work with W. Dorabiala.

September 26th Ilker Yuce SUNY Oswego Decompositions of 2-generator free Kleinian groups and hyperbolic displacements Let H³ denotes the hyperbolic 3-space. An orientable hyperbolic 3-manifold M may be viewed as the quotient H³/Γ where Γ is a discrete group of orientation preserving isometries of H³. Let Γ be a discrete torsion-free subgroup of orientation preserving isometries of H³ generated by the elements ξ and η. A positive real number λ is called a Margulis number for the group Γ or the orientable hyperbolic 3 manifold M if the inequality max{dist(z,ξ• z),dist(z, η• z)}≥λ holds for every point z ∈ H³.
The largest known Margulis number for a closed orientable hyperbolic 3-manifold whose fundamental group has no 2-generator subgroup of finite index is log3≈ 1.098 obtained by Culler and Shalen. As a consequence, every such hyperbolic 3-manifold contains a hyperbolic ball of radius ½ log 3 and has a volume greater than 0.92. Culler, Hersonsky and Shalen improve the previous volume estimate to 0.94. The largest known lower bound for the volume of these manifolds is 1.1 obtained by Przeworski . The Log 3 Theorem, which introduces the Margulis number log 3, lies in the foundation of all of these volume estimates:

The Log 3 Theorem.(Culler-Shalen) Let ξ and η be non-commuting isometries of H³. Suppose that ξ and η generate a torsion-free, discrete group which is topologically tame, is not co-compact and contains no parabolics. Then every point of the hyperbolic space is moved a distance at least log 3 by either ξ or η.
In my talk, I'll prove a refinement of the Log 3 Theorem and a generalization of it by introducing a process that is likely to give improved Margulis numbers and hence improved lower volume estimates for closed orientable hyperbolic 3-manifolds whose fundamental group has no 2-generator subgroup of finite index.

October 3rd Bill Menasco University at Buffalo On a theorem of Hass, Thompson and Thurston: Stabilization of Heegaard Splittings

October 10th Richard H. Escobales, Jr. Canisius College Foliations by Minimal Surfaces and Ricci Curvature Let (Mⁿ,g) be a closed, connected, oriented, smooth, Riemannian, n-manifold with a transversely oriented, codimension-2 foliation F. Suppose the transverse volume form m is basic and {X,Y } are basic vector fields, so m(X,Y) = 1. Then the leaf component of [X,Y], V[X,Y], is globally defined on M and is independent of the basic pair of vector fields {X,Y } satisfying the above equation as observed by Cairns in [C]. Using the Bochner technique, we show under appropriate assumptions on cohomology and on the Ricci curvature of the leaves of the foliation F, that the distribution orthogonal to that of the leaves, H, is integrable and the leaves of this new foliation are minimal surfaces of M. In the second section we provide some analogous results for the special case when F is a Riemannian foliation.

October 17th Diane Vavrichek SUNY Binghamton Quasi-isometry invariant commensurizer subgroups I will present results about the quasi-isometry invariance of the existence and location of certain infinite cyclic subgroups and their commensurizers in one-ended finitely presented groups. An application of this is the quasi-isometry invariance of certain vertex groups of the Scott-Swarup JSJ decomposition for groups.

October 24th Will Wylie University of Pennsylvania TBA

October 31st

November 7th

November 14th

November 31st

December 5th

Old seminar pages:

Spring 2007

Fall 2007

Spring 2008

Date	Speaker	Organization	Title	Abstract
September 5th	Organizational Meeting
September 12th	Chris Leininger	U.I.U.C.	Flat structures via length functions	I'll discuss joint work with M. Duchin and K. Rafi in which we study flat structures on a closed surface S using lengths of closed geodesics on S.
September 19th	Bernard Badzioch	University at Buffalo	Localizations of mapping spaces.	Homotopy localizations are functors which can be interpreted as projections of the category of topological spaces onto its subcategory. This class of functors includes Postnikov sections, homological localizations, Quillen plus construction etc. One of the central questions one encounters while dealing with such functors is which topological properties of spaces they preserve. The talk will discuss this problem in relation to preservation of the structure of mapping spaces Map(A, X) for a given space A. This is a joint work with W. Dorabiala.
September 26th	Ilker Yuce	SUNY Oswego	Decompositions of 2-generator free Kleinian groups and hyperbolic displacements	Let H³ denotes the hyperbolic 3-space. An orientable hyperbolic 3-manifold M may be viewed as the quotient H³/Γ where Γ is a discrete group of orientation preserving isometries of H³. Let Γ be a discrete torsion-free subgroup of orientation preserving isometries of H³ generated by the elements ξ and η. A positive real number λ is called a Margulis number for the group Γ or the orientable hyperbolic 3 manifold M if the inequality max{dist(z,ξ• z),dist(z, η• z)}≥λ holds for every point z ∈ H³. The largest known Margulis number for a closed orientable hyperbolic 3-manifold whose fundamental group has no 2-generator subgroup of finite index is log3≈ 1.098 obtained by Culler and Shalen. As a consequence, every such hyperbolic 3-manifold contains a hyperbolic ball of radius ½ log 3 and has a volume greater than 0.92. Culler, Hersonsky and Shalen improve the previous volume estimate to 0.94. The largest known lower bound for the volume of these manifolds is 1.1 obtained by Przeworski . The Log 3 Theorem, which introduces the Margulis number log 3, lies in the foundation of all of these volume estimates: The Log 3 Theorem.(Culler-Shalen) Let ξ and η be non-commuting isometries of H³. Suppose that ξ and η generate a torsion-free, discrete group which is topologically tame, is not co-compact and contains no parabolics. Then every point of the hyperbolic space is moved a distance at least log 3 by either ξ or η. In my talk, I'll prove a refinement of the Log 3 Theorem and a generalization of it by introducing a process that is likely to give improved Margulis numbers and hence improved lower volume estimates for closed orientable hyperbolic 3-manifolds whose fundamental group has no 2-generator subgroup of finite index.
October 3rd	Bill Menasco	University at Buffalo	On a theorem of Hass, Thompson and Thurston: Stabilization of Heegaard Splittings
October 10th	Richard H. Escobales, Jr.	Canisius College	Foliations by Minimal Surfaces and Ricci Curvature	Let (Mⁿ,g) be a closed, connected, oriented, smooth, Riemannian, n-manifold with a transversely oriented, codimension-2 foliation F. Suppose the transverse volume form m is basic and {X,Y } are basic vector fields, so m(X,Y) = 1. Then the leaf component of [X,Y], V[X,Y], is globally defined on M and is independent of the basic pair of vector fields {X,Y } satisfying the above equation as observed by Cairns in [C]. Using the Bochner technique, we show under appropriate assumptions on cohomology and on the Ricci curvature of the leaves of the foliation F, that the distribution orthogonal to that of the leaves, H, is integrable and the leaves of this new foliation are minimal surfaces of M. In the second section we provide some analogous results for the special case when F is a Riemannian foliation.
October 17th	Diane Vavrichek	SUNY Binghamton	Quasi-isometry invariant commensurizer subgroups	I will present results about the quasi-isometry invariance of the existence and location of certain infinite cyclic subgroups and their commensurizers in one-ended finitely presented groups. An application of this is the quasi-isometry invariance of certain vertex groups of the Scott-Swarup JSJ decomposition for groups.
October 24th	Will Wylie	University of Pennsylvania	TBA
October 31st
November 7th
November 14th
November 31st
December 5th