August 29- Cockins Hall 240, 3:30 pm-4:30 pm
Rita Gitik (University of Michigan)
On Geodesic Triangles in the Hyperbolic Plane
Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.
September 5- Enarson Classroom Building 0206, 1:50 pm-2:50 pm
Rachel Skipper (OSU)
Finiteness Properties for Simple Groups
A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\"{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$. As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups.
September 19- MA 317, 1:50 pm-2:50 pm
David Constantine (Wesleyan University)
Markov codings for geodesic flow on CAT(-1) spaces
In this talk I'll discuss some joint work with Jean-Francois Lafont and Dan Thompson constructing a strong Markov coding for the geodesic flow on a CAT(-1) space. This has a number of important dynamical consequences, but I'll focus on the geometric underpinnings of the result -- how the CAT(-1) curvature property makes the argument run. If time permits, I hope to discuss an application of this result to the co-amenability problem for isometry groups of CAT(-1) spaces. This application gives an alternate proof of a result of Coulon, Dal'Bo, and Sambusetti.
September 24- CH 240, 3:00 pm-4:00 pm
Yvon Verberne (University of Toronto)
Constructing pseudo-Anosov homeomorphisms using positive twists
Thurston obtained the first construction of pseudo-Anosov homeomorphisms by showing the product of two parabolic elements is pseudo-Anosov. A related construction by Penner involved constructing whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves. In this talk, I will present a new construction of pseudo-Anosov homeomorphisms and discuss some of the unique properties associated to maps obtained by this new construction.
October 1- CH 240, 3:00 pm-4:00 pm
Luis Jorge Sánchez Saldaña (OSU)
The Eilenberg-Ganea problem and groups acting on trees
In this talk we will define geometric and cohomological dimension for groups relative to a family of subgroups. Next, we will enunciate an Eilenberg-Ganea type theorem in this context, proved y W. Lück and D. Meintrup. This theorem claims that both dimensions are equal if the cohomological dimension is at least 3 or the geometric dimension is at least 4. The Eilenberg-Ganea problem for families is the question of whether both dimensions always coincide. N. Brady, I. Leary, and B. Nucinkis constructed examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the family of finite subgroups. We will discuss how to construct more examples using known examples of groups with cohomological dimension 2 and geometric dimension 3 for a given family, and Bass-Serre theory.
October 8- CH 240, 3:00 pm-4:00 pm
Mark Pengitore (OSU)
Effective conjugacy separability of wreath products
In this talk, I will discuss joint work with Michal Ferov where we characterize C-conjugacy separability of a wreath product of C-conjugacy separable groups where C is an extension-closed psuedovariety of finite groups. As a consequence of our methods, we provide asymptotic bounds for conjugacy separability of wreath products of nilpotent groups which include the lamplighter groups. As a final application, we provide asymptotic bounds for conjugacy separability of the free metabelian group of rank m.
October 15- CH 240, 3:00 pm-4:00 pm
Chris Leininger (University of Illinois Urbana-Champaign)
Weil-Petersson translation length
In this talk, I will discuss joint work with Minsky, Souto, and Taylor in which we prove that any mapping torus of a pseduo-Anosov mapping class with bounded normalized Weil-Petersson (WP) translation length contains a finite set of "vertical and horizontal closed curves", and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds (depending only on the normalized WP length bound). This echoes an earlier result, joint with Farb and Margalit, for the Teichmuller metric. We also prove new estimates for the WP translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.
October 17-MA 317, 1:50 pm-2:50pm
Michael Ferov (The University of Newcastle)
Linguistic complexity of primitive sets in free groups
Abstract: A set of elements of a free group is said to be primitive if it can be extended to a free base. In my talk I will sketch a proof that, as a formal language, the set of primitive sets of a free group is context sensitive. Time permitting, I will discuss that in the case of the free group on two generators the linguistic complexity of primitives can be brought down to EDT0L. No background in former languages is assumed - I will give all the necessary definitions.
October 29-CH 240, 3:00 pm-4:00 pm
Daniel Studenmund (University of Notre Dame)
Algebra and geometry of finite-index subgroups
Abstract: Given an infinite, discrete group G, we will discuss algebraic and geometric structures on the collection C(G) of its finite-index subgroups. The abstract commensurator of G, Comm(G), is an algebraic structure associated to C(G) that can detect surprising data about G. We will discuss some known results and pose questions about Comm(F_2). We then define a metric space structure on C(G) and discuss results in subgroup growth, and use this to motivate the more general notion of commensurability growth. This talk includes discussion of work with Khalid Bou-Rabee, Tasho Kaletha, and Rachel Skipper.
November 5-CH 240, 3:00 pm-4:00 pm
Edgar A. Bering IV (Temple University)
Special covers of alternating links
The “virtual conjectures” in low-dimensional topology, stated by Thurston in 1982, postulated that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved in 2012 by Agol and Wise, using the machine of special cube complexes. Since that time, many mathematicians have asked how big a cover one needs to take to ensure one of these desired properties.
We begin to give a quantitative answer to this question, in the setting of alternating links in S^3. If a prime alternating link L has a diagram with n crossings, we prove that the complement of L has a special cover of degree less than 72((n-1)!)^2. As a corollary, we bound the degree of the cover required to get Betti number at least k. We also quantify residual finiteness, bounding the degree of a cover where a closed curve of length k fails to lift. This is joint work with David Futer.
November 7-MA 317, 1:50 pm-2:50pm
Noelle Sawyer (Wesleyan University)
Partial marked length spectrum rigidity for negatively curved surfaces
The marked length spectrum of a metric on a compact Riemannian manifold records the length of the shortest closed curve in each free homotopy class. It is known that a negatively curved metric on a compact Riemannian manifold is uniquely determined by its marked length spectrum up to isometry. My results show that under certain conditions on the excluded homotopy classes, a partial marked length spectrum also uniquely determines such a metric.
November 12-CH 240, 3:00 pm-4:00 pm
Jacob Russell (City University of New York)
The geometry of subgroup combination theorems
While producing subgroups of a group by specifying generators is easy, understanding the structure of such a subgroup is notoriously difficult problem. In the case of hyperbolic groups, Gitik utilized a local-to-global property to produce an elegant condition that ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that the mapping class group of a surface and many other topologically important groups have a similar local-to-global property from which an analogy of Gitik's result can be obtained. In the case of the mapping class group, this produces a combination theorem for the dynamically and geometrically important convex cocompact subgroups. Joint work with Davide Spriano and Hung C. Tran.
November 12-BE0394 4:10 pm-5:10 pm
Sean Cleary (City College of New York)
Deep dead ends in finitely presented groups
A dead end in the Cayley graph of a finitely generated group with respect to a particular generating set is an element beyond which a geodesic ray in the Cayley graph from the identity cannot be extended. Dead ends occur in a variety of settings, and occur in different levels of severity, measured by the depth of a dead end. I will describe some aspects of dead end phenonema in several families of groups, including an interesting finitely-presented metabelian group constructed by Baumslag. This last example is a finitely-presented group with unbounded dead-end depth, shown in joint work with Tim Riley.
November 25-SOE0241 3:00 pm-4:00 pm
Camille Horbez (Université Paris Sud)
Algebraic rigidity for Out(Fn)
Let n>3, let Fn be a free group of rank n, and let Out(Fn) be its outer automorphism group. Farb and Handel proved in 2007 that every isomorphism between two finite-index subgroups of Out(Fn) coincides with the conjugation by an element of the ambient group Out(Fn). This can be viewed as an analogue for Out(Fn) to the Mostow rigidity theorem. I will present a new proof of the Farb-Handel theorem, which enables us to extend it in various directions, and in particular compute the `symmetries' of many interesting subgroups of Out(Fn). This talk is based on joint works with Ric Wade, and with Sebastian Hensel and Ric Wade.
December 3-CH 240 3:00 pm-4:00 pm
Jingyin Huang (OSU)
The Helly geometry of Artin groups and Garside groups
Garside groups and Artin groups are two generalizations of braid groups. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, hence equip these groups with a particular nonpositive-curvature-like (NPC) structure. Such structure shares many properties of a CAT(0) structure and has some additional combinatorial flavor. We shall explain this NPC structure in more detail and discuss new results on the topology and geometry of these groups which are immediate consequences of such structure. This is joint work with D. Osajda.