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Martin Lorenz, Temple University
This series of three talks will deal with "growth" of groups and ofalgebras. Despite its elementary combinatorially flavored definition,the concept of growth has played in important role in algebra andother areas; in fact, for groups, its origins lie in geometry and themain theorems have been contributed by geometers. Certain theoremsabout groups become “easy” when viewed in the context ofalgebras. The talks aim to explore the potential and the currentlimitations of this approach.
Jack Hanson, City College of NY, CUNY
In their study of percolation, physicists have proposed "scaling hypotheses" relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ — the probability that the open cluster of the origin has Euclidean diameter at least $n$.
As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.
Khánh Lê, Temple University
Abstract: A group is called left-orderable if it admits a total ordering that is invariant under left multiplication. In 3-manifold topology, left orderability is an important concept due to its role in the L-space conjecture. There has been a substantial effort in developing tools to order the fundamental group of rational homology 3-spheres. In a recent work, Xinghua Gao encoded information about hyperbolic \(\widetilde{PSL}_2{\mathbb R}\) representations of a one-cusped 3-manifold \(M\) in the holonomy extension locus and used it to order intervals of Dehn fillings assuming a strong technical condition of the character variety of \(M\). In this talk, we will show how to weaken this condition to a local condition at the non-abelian reducible representation. As an application, we construct left orders on an interval of Dehn fillings on the \([1,1,2,2,2j]\) two-bridge knots.
There are no conferences this week.