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Spring Semester 2016

Date / Time Speaker Title Location
10 February 2016
15:45-16:45 		Benjamin Beeker
The Hebrew University of Jerusalem
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Geometry Seminar

Title Resolutions of CAT(0) cube complex and accessibility properties
Speaker, Affiliation Benjamin Beeker, The Hebrew University of Jerusalem
Date, Time 10 February 2016, 15:45-16:45
Location HG G 43
Abstract In this joint work with Nir Lazarovich, we generalize the idea of resolutions defined by Dunwoody for trees to higher dimensional CAT(0) cube complexes. In dimension two, we give a bound for the number of hyperplanes of this resolution. This result has applications for surfaces and 3-manifolds. In this talk, I will discuss these applications as well as describe Dunwoody's construction and how to extend it to higher dimensions.
Resolutions of CAT(0) cube complex and accessibility propertiesread_more
HG G 43
2 March 2016
15:45-16:45 		Colin Reid
University of Newcastle, Australia
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Geometry Seminar

Title Chief series in locally compact groups
Speaker, Affiliation Colin Reid, University of Newcastle, Australia
Date, Time 2 March 2016, 15:45-16:45
Location HG G 43
Abstract I will be talking about joint work with Phillip Wesolek. A chief factor of a topological group $G$ is a factor $K/L$, where $K$ and $L$ are closed normal subgroups such that no closed normal subgroup of $G$ lies strictly between $K$ and $L$. In general it is not possible to break up a discrete group into chief factors, so the same is true of locally compact groups. However, we can show that a compactly generated locally compact group admits an 'essentially chief series', that is, a finite normal series in which each of the factors is compact, discrete or a chief factor. There is also a sense in which the essentially chief series is 'unique up to equivalence', however one must allow for the fact that a product of two closed normal subgroups is not always closed.
Chief series in locally compact groupsread_more
HG G 43
9 March 2016
15:45-16:45 		Ilya Khayutin
The Hebrew University of Jerusalem
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Geometry Seminar

Title Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits
Speaker, Affiliation Ilya Khayutin, The Hebrew University of Jerusalem
Date, Time 9 March 2016, 15:45-16:45
Location HG G 43
Abstract In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of PGLn and SLn. These invariants allow us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split. Packets of periodic torus orbits are natural collections of torus orbits coming from a single rational adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The distribution of packets of periodic torus orbit has been studied using dynamical methods in the pioneering work of Linnik in the rank 1 case (equidistribution on the 2-sphere) and by Einsiedler, Lindenstrauss, Michel and Venkatesh (ELMV) in higher rank. We note that in rank 1, stronger equidistribution results for packets of periodic orbits have been established by Duke and Iwaniec using the theory of automorphic functions. The dynamical approach typically consists of two main ingredients: an arithmetic one, which implies that the toral packets have high asymptotic metric entropy, and a measure rigidity argument, which deduces from the entropy result a statement regarding the limit distribution of the orbits. While thanks to Einsiedler, Katok and Lindenstrauss we have very powerful measure rigidity tools for higher rank toral actions, we know much less regarding the arithmetic of these packets in higher rank. A notable exception is the work of ELMV which synergies the dynamical approach with harmonic analysis to prove an equidistribution theorem similar to Linnik’s in the split rank 2 case. In the other cases the current known results, due to the same authors, are significantly weaker. Our methods generalize Linnik's original arithmetic approach in a different direction. We derive new invariants, akin to the discriminant inner product used by Linnik. These invariants come from studying double quotients of a reductive group by a torus using geometric invariant theory. We then derive a sharper lower bound for the entropy from these invariants using the action of the Galois group of the torus’ splitting field and the algebraic relations between the invariants. This lower bound gives new qualitative restrictions on the possible limit measures and applies also to partially split maximal tori.
Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbitsread_more
HG G 43
23 March 2016
15:45-16:45 		Anne Thomas
University of Sydney and FIM
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Geometry Seminar

Title Affine Deligne-Lusztig varieties and the geometry of Euclidean reflection groups
Speaker, Affiliation Anne Thomas, University of Sydney and FIM
Date, Time 23 March 2016, 15:45-16:45
Location HG G 43
Abstract Let G be a reductive group such as $SL_n$ over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the affine Weyl group of G(F). The associated affine Deligne-Lusztig varieties X_x(b) were introduced by Rapoport. These are indexed by elements x in W and b in G(F), and are related to many important concepts in algebraic geometry over fields of positive characteristic. Basic questions about the varieties X_x(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. For these questions, it suffices to consider elements x and b both in W. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that b is a translation. Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns. Since we work only in the standard apartment of the affine building for G(F), which is just the tessellation of Euclidean space induced by the action of the reflection group W, our results also hold over the p-adics. We obtain an application to reflection length in W. This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).
Affine Deligne-Lusztig varieties and the geometry of Euclidean reflection groupsread_more
HG G 43
6 April 2016
15:45-16:45 		Alina Vdovina
Newcastle University and FIM
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Geometry Seminar

Title Expanders, buildings and Beauville surfaces
Speaker, Affiliation Alina Vdovina, Newcastle University and FIM
Date, Time 6 April 2016, 15:45-16:45
Location HG G 43
Abstract We'll present buildings as universal covers of certain CAT(0) complexes. Fundamental groups of these complexes will be used for expander constructions and for generating an infinite families of Beauville surfaces.
Expanders, buildings and Beauville surfacesread_more
HG G 43
13 April 2016
15:45-16:45 		Alexander Lubotzky
Hebrew University and ETH-ITS
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Geometry Seminar

Title Finite transformation groups of locally symmetric spaces: on two problems of Borel
Speaker, Affiliation Alexander Lubotzky, Hebrew University and ETH-ITS
Date, Time 13 April 2016, 15:45-16:45
Location HG G 43
Abstract Let M be a locally symmetric irreducible closed manifold of dimension ≥ 3. A result of Borel combined with Mostow rigidity imply that there exists a finite group G = G(M) such that any finite subgroup of Homeo+(M) is isomorphic to a subgroup of G. Borel asked if there exist M’s with G(M) trivial and if the number of conjugacy classes of finite subgroups of Homeo+(M) is finite. We answer both questions: (1) For every finite group G there exist M’s with G(M) = G, and (2) the number of maximal subgroups of Homeo+(M) can be either one, countably many or continuum and we determine (at least for dim M not equal 4) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces. Joint work with Sylvain Cappell and Shmuel Weinberger.
Finite transformation groups of locally symmetric spaces: on two problems of Borelread_more
HG G 43
20 April 2016
15:45-16:45 		Damian Osajda
Polish Academy of Sciences & University of Wroclaw
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Geometry Seminar

Title Groups containing expanders
Speaker, Affiliation Damian Osajda, Polish Academy of Sciences & University of Wroclaw
Date, Time 20 April 2016, 15:45-16:45
Location HG G 43
Abstract I will present my recent construction of finitely generated groups containing isometrically embedded expanders. Such groups have many exotic properties. For instance, they do not embed coarsely into a Hilbert space, and the Baum-Connes conjecture with coefficients fails for them. The construction allows us to provide the first examples of groups that lack property A (i.e., they are not exact) but are still coarsely embeddable into a Hilbert space. Better still, these groups act properly on CAT(0) cubical complexes. To end with, I will also present some further applications of the main construction concerning aspherical manifolds and the asymptotic dimension.
Groups containing expandersread_more
HG G 43
4 May 2016
15:45-16:45 		Felipe Riquelme
Université de Rennes 1
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Geometry Seminar

Title Escape of mass and entropy for the geodesic flow in negative curvature
Speaker, Affiliation Felipe Riquelme, Université de Rennes 1
Date, Time 4 May 2016, 15:45-16:45
Location HG G 43
Abstract In this talk we describe some properties relating the escape of mass of a sequence of probability measures on the unit tangent bundle of a generalized Schottky manifold and the measure-theoretic entropy of the geodesic flow. Roughly speaking, we prove that a sequence of measures with large entropy cannot lose the whole mass. Moreover, the critical entropy value needed to ensure a remaining of mass is related to the geometry of the noncompact part of the manifold. Our methods are based on the one side on the symbolic realisation of the geodesic flow, and on the other, on geometric techniques and approximation properties at level of groups. This is a joint work with Godofredo Iommi and Anibal Velozo.
Escape of mass and entropy for the geodesic flow in negative curvatureread_more
HG G 43
11 May 2016
15:45-16:45 		Arie Levit
Weizmann Institute of Science, Israel
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Geometry Seminar

Title Local rigidity in locally compact groups
Speaker, Affiliation Arie Levit, Weizmann Institute of Science, Israel
Date, Time 11 May 2016, 15:45-16:45
Location HG G 43
Abstract A lattice is topologically locally rigid (t.l.r) if small deformations of it are isomorphic lattices. Uniform lattices in Lie groups were shown to be t.l.r by Weil [60']. We show that uniform lattices are t.l.r in any topological group G, given the assumption that G is compactly generated. A lattice is locally rigid (l.r) is small deformations arise from conjugation. It is a classical fact due to Weil [62'] that lattices in semi-simple Lie groups are l.r. Relying on our t.l.r results and on recent work by Caprace-Monod we prove l.r for uniform lattices in the isometry groups of proper geodesically complete CAT(0) spaces, with the exception of SL2(R) factors that occurs already in the classical case. Moreover we are able to extend certain finiteness results due to Wang to this more general context. In my talk I will explain the above notions and results, and present some ideas from the proofs. This is a joint work with Tsachik Geladner.
Local rigidity in locally compact groupsread_more
HG G 43
18 May 2016
15:45-16:45 		Michelle Bucher
Université de Genève
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Geometry Seminar

Title On the minimal growth rates of amalgamated products and HNN-extensions
Speaker, Affiliation Michelle Bucher, Université de Genève
Date, Time 18 May 2016, 15:45-16:45
Location HG G 43
Abstract The minimal growth rate of a finitely generated group G is the infimum over all finite generating sets S of G of the growth rate of G with respect to S. Motivated by questions by Avinoam Mann, I will discuss geometric tools to compute or estimate the minimal growth rates of certain families of groups, in particular for the metabelian Baumslag-Solitar groups. Joint work with Alexey Talambutsa.
On the minimal growth rates of amalgamated products and HNN-extensionsread_more
HG G 43
25 May 2016
15:45-16:45 		Aditi Kar
University of Southampton
Details

Geometry Seminar

Title Ping Pong in CAT(0) cube complexes
Speaker, Affiliation Aditi Kar, University of Southampton
Date, Time 25 May 2016, 15:45-16:45
Location HG G 43
Abstract Groups acting on CAT(0) cube complexes have played a seminal role in contemporary mathematics. I am interested in properties of CAT(0) cubical groups like property P_naive and uniform exponential growth. Property P_naive is a very strong form of the Tits Alternative and was introduced by de la Harpe to study the analytic property of a group being C*-simple. Variations on this theme lead to a concrete description of free sub-semigroups contained inside groups acting on CAT(0) cube complexes of a given dimension. This is joint work with Michah Sageev (Technion) and leads us to interesting variations of uniform exponential growth.
Ping Pong in CAT(0) cube complexesread_more
HG G 43
1 June 2016
15:45-16:45 		Alina Vdovina
Newcastle University and FIM
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Geometry Seminar

Title Superrigidity and maps of surfaces into hyperbolic buildings
Speaker, Affiliation Alina Vdovina, Newcastle University and FIM
Date, Time 1 June 2016, 15:45-16:45
Location HG G 43
Abstract One of the most well known open questions in geometric group theory is the following question of Gromov: is it true, that every one-ended hyperbolic group contains a surfaces group? Even if we restrict ourselves to groups acting on hyperbolic buildings the answer is not known in general. We'll present some non-obvious embeddings of surfaces groups into groups acting on buildings as well as some negative results, leaving a possibility for counter-examples.
Superrigidity and maps of surfaces into hyperbolic buildingsread_more
HG G 43

Organisers: Marc Burger, Manfred Einsiedler, Alessandra Iozzi, Urs Lang, Viktor Schröder, Alessandro Sisto

Archive: AS 24 SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09 SS 09

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