Past Algebra Seminar

17 June 2014
17:00
Prof. Travis Schedler.
Abstract
The Springer Correspondence relates irreducible representations of the Weyl group of a semisimple complex Lie algebra to the geometry of the cone of nilpotent elements of the Lie algebra. The zeroth Poisson homology of a variety is the quotient of all functions by those spanned by Poisson brackets of functions. I will explain a conjecture with Proudfoot, based on a conjecture of Lusztig, that assigns a grading to the irreducible representations of the Weyl group via the Poisson homology of the nilpotent cone. This conjecture is a kind of symplectic duality between this nilpotent cone and that of the Langlands dual. An analogous statement for hypertoric varieties is a theorem, which relates a hypertoric variety with its Gale dual, and assigns a second grading to its de Rham cohomology, which turns out to coincide with a different grading of Denham using the combinatorial Laplacian.
10 June 2014
17:00
Jon Gonzalez Sanchez
Abstract
Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G is called p-nilpotent if S has a normal complement N in G, that is, G is the semidirect product between S and N. The notion of p-nilpotency plays an important role in finite group theory. For instance, Thompson's criterion for p-nilpotency leads to the important structural result that finite groups with fixed-point-free automorphisms are nilpotent. By a classical result of Tate one can detect p-nilpotency using mod p cohomology in dimension 1: the group G is p-nilpotent if and only if the restriction map in cohomology from G to S is an isomorphism in dimension 1. In this talk we will discuss cohomological criteria for p-nilpotency by Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the 1960s and 1970s. Finally, we will discuss how one can extend Tate's result to study p-solvable and more general finite groups.
3 June 2014
17:00
Tsachik Gelander
Abstract
Gromov and Piatetski-Shapiro proved the existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about $v^v$ such manifolds of volume at most $v$, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi- isometry classes of lattices in $SO(n,1)$. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms. A joint work with Arie Levit.
27 May 2014
17:00
Michael Collins
Abstract
In 1878, Jordan showed that if $G$ is a finite group of complex $n \times n$ matrices, then $G$ has a normal subgroup whose index in $G$ is bounded by a function of $n$ alone. He showed only existence, and early actual bounds on this index were far from optimal. In 1985, Weisfeiler used the classification of finite simple groups to obtain far better bounds, but his work remained incomplete when he disappeared. About eight years ago, I obtained the optimal bounds, and this work has now been extended to subgroups of all (complex) classical groups. I will discuss this topic at a “colloquium” level – i.e., only a rudimentary knowledge of finite group theory will be assumed.
27 May 2014
15:00
Dennis Dreesen
Abstract
The common convention when dealing with hyperbolic groups is that such groups are finitely generated and equipped with the word length metric relative to a fi nite symmetric generating subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the fi nitely generated setting. We study the class of locally compact hyperbolic groups and elaborate on the similarities and diff erences between the discrete and non-discrete setting.
13 May 2014
17:00
Eric Swenson
Abstract
Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and $f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$. We prove that the join of two Cantor sets and its suspension are Tits rigid.
6 May 2014
17:00
Alain Valette
Abstract
A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.
29 April 2014
17:00
Abstract
A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$ is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a so called state-closed (or, self-similar in dynamical terms) representation of $G$ on a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$; i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant. Examples of state-closed groups are the Grigorchuk 2-group and the Gupta- Sidki $p$-groups in their natural representations on rooted trees. The affine group $Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed representations. Yet another example is the free nilpotent group $G = F (c; d)$ of class c, freely generated by $x_i (1\leq  i \leq  d)$: let $H = \langle x_i^n | \ (1 \leq  i \leq  d) \rangle$ where $n$ is a fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i \rightarrow x_i$ $(1 \leq  i \leq  d)$. We will discuss state-closed representations of general abelian groups and of …nitely generated torsion-free nilpotent groups.

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