Fall 2021 – Now in hybrid.

Abstract: A main feature of the theory of right-angled Artin groups (RAAGs) consists in the fact that the algebraic properties of the group can be described in terms of the combinatorial properties of its defining graph. This idea carries over to the study of Artin kernels, i.e. subgroups of RAAGs obtained as kernels of maps to the integers. For some specific classes of chordal graphs, we obtain a sharp structural dichotomy for the Artin kernels. We will discuss some applications to the study of splittings, fibrations, and BNS invariants of these groups. This talk is based on joint work with M. Barquinero and K. Ye, and joint work with Y.-C. Chang.

Abstract: Growth of finitely generated groups studies the cardinality of balls as the radius grows.   While precise growth rates are generating set dependent, it is sometimes possible to uniformly control the growth rates of all generating sets.  I will discuss both geometric and algebraic tools that relate subgroup and quotient structure of a group to bounding growth rates.  Using these ideas, I will discuss joint work with Robert Kropholler and Rylee Lyman proving an exponential growth gap for subgroups generated by automorphisms of one-ended hyperbolic groups.

Abstract: Bowditch associated a topological space ∂G to Relatively Hyperbolic Group G. Topological information about ∂G is often useful to understand algebraic information about the group G. In this talk we will discuss the background and also sketch a proof of the following theorem that resolves a 20-year old open question in. Geometric Group Theory:
If G is a relatively one-ended, relatively hyperbolic group then ∂G is locally connected.
Earlier Bowditch proved the local connectedness of ∂G with a more restricted hypothesis. We will sketch the proof of a more general result.

Abstract: The study of embedded surfaces in hyperbolic 3-manifolds has led to several major advances in the fields of geometry, topology, and geometric group theory. In this talk we address the higher dimensional analogue of embedded 3-manifolds in hyperbolic 4-manifolds. In particular, we address the existence of totally geodesic 3-manifolds in small volume hyperbolic 4-manifolds.

Abstract: A fundamental question in low-dimensional topology asks which groups can arise as subgroups of the diffeomorphism group, homeomorphism group, and mapping class group of a surface. A classical approach to this problem is to find a group which acts by isometries on the surface, implying that this group is a subgroup of all three groups above. In this talk, I will describe a new and different approach to finding subgroups of (big) mapping class groups which does not rely on constructing isometries of the surface. In particular, the subgroups we construct will not be contained in the isometry group of the surface, and, moreover, the subgroups will be “large” in a sense I will make precise. Finally, I will describe a combination theorem for indicable subgroups of big mapping class groups, one application of which is a new construction of right-angled Artin groups in big mapping class groups. This is joint work with Hannah Hoganson, Marissa Loving, Priyam Patel, and Rachel Skipper.

Abstract: An n-dimensional  Kleinian group is a torsion-free, discrete isometry subgroup of the n-dimensional hyperbolic space. In this talk, we will introduce a new way to construct higher dimensional Kleinian groups from right-angled Artin groups via the strict hyperbolization introduced by Charney and Davis. In particular, we will focus on the construction from the direct product of n copies of the rank 2 free group.

Abstract: This talk is about hyperbolicity of surface group extensions and a question of Farb-Mosher about whether purely pseudo-Anosov subgroups of mapping class groups are convex cocompact. I will explain this problem and give an answer for subgroups of the genus-2 Goeritz group, which is the group of mapping classes of a genus-2 surface that extend to the genus-2 Heegaard splitting of the 3-sphere.

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Abstract: A group is said to “virtually algebraically fiber” if it has a finite index subgroup admitting a map onto Z with finitely generated kernel. Stronger than finite generation, if the kernel is even of type F_n for some n then we say the group “virtually algebraically F_n-fibers”. Right-angled Coxeter groups (RACGs) are a class of groups for which the question of virtual algebraic F_n-fibering is of great interest. In joint work with Eduard Schesler, we introduce a new probabilistic criterion for the defining flag complex that ensures a RACG virtually algebraically F_n-fibers. This expands on work of Jankiewicz–Norin–Wise, who developed a way of applying Bestvina–Brady Morse theory to the Davis complex of a RACG to deduce virtual algebraic fibering. We apply our criterion to the special case where the defining flag complex comes from a certain family of finite buildings, and establish virtual algebraic F_n-fibering for such RACGs. The bulk of the work involves proving that a “random” (in some sense) subcomplex of such a building is highly connected, which is interesting in its own right.

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