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Autumn Semester 2017

Date / Time Speaker Title Location
27 September 2017
15:45-16:45 		Prof. Dr. Peter Feller
ETH Zurich, Switzerland
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Geometry Seminar

Title The fractional Dehn twist coefficient of braids and homogenizations of knot invariants
Speaker, Affiliation Prof. Dr. Peter Feller, ETH Zurich, Switzerland
Date, Time 27 September 2017, 15:45-16:45
Location HG G 43
Abstract We discuss connections between two different points of view on Artin's braid groups: braids as mapping classes of punctured discs and braids as a tool to study knots via the closure operation. Our main result is a characterization of the `fractional Dehn twist coefficient' of braids — a rational number associated with a mapping class first introduced by Gabai and Oertel — in terms of knot invariants due to Ozsvath, Stipsicz, and Szabo. We provide consequences about the complexity of braids (and knots that arise as their closure) for braids with `many twists'. A key ingredient for the characterization are homogeneous quasi-morphisms on braid groups that arise from knot invariants.
The fractional Dehn twist coefficient of braids and homogenizations of knot invariantsread_more
HG G 43
4 October 2017
15:45-16:45 		Yash Lodha
EPFL
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Geometry Seminar

Title Chain groups of homeomorphisms of the interval and the circle
Speaker, Affiliation Yash Lodha, EPFL
Date, Time 4 October 2017, 15:45-16:45
Location HG G 43
Abstract We introduce the notion of chain groups of homeomorphisms of a one-manifolds, which are groups finitely generated by homeomorphisms, each supported on exactly one interval in a chain, subject to a certain mild dynamical condition. The resulting class of groups exhibits a combination of uniformity and diversity of properties. We distinguish isomorphism types, study normal subgroups structure and actions by various degrees of regularity. This represents work in two articles. The first is joint work with Sang-hyun Kim and Thomas Koberda and the second is joint work with Thomas Koberda.
Chain groups of homeomorphisms of the interval and the circleread_more
HG G 43
18 October 2017
15:45-16:45 		Corina Ciobotaru
University of Fribourg
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Geometry Seminar

Title Cartan limits of SL(n, Q_p)
Speaker, Affiliation Corina Ciobotaru, University of Fribourg
Date, Time 18 October 2017, 15:45-16:45
Location HG G 43
Abstract For a locally compact group G the set of all its closed subgroups S(G) is endowed with the Chabauty topology, under which S(G) becomes a compact space. Given a family of closed subgroups of G satisfying some properties it is then natural to ask if its limit subgroups in S(G) preserve the same properties and if we can explicitly compute them.
In a recent joint work with Arielle Leitner we study the limits under the Chabauty topology of Cartan subgroups of SL(n,Q_p), the closed subgroups SL(n, Q_p)-conjugated to the diagonal subgroup of SL(n, Q_p). When a limit subgroup contains only elliptic elements we prove that up to conjugacy it is contained in the unipotent radical of the Borel subgroup of SL(n,Q_p). The key idea is an explicit homeomorphism between the Chabauty closure of Cartan sub-algebras of sl(n, Q_p) and the Chabauty closure of Cartan subgroups of SL(n,Q_p). By the Flat Torus Theorem when a limit subgroup contains hyperbolic elements we prove it preserves a flat, not necessary of maximal dimension, in the Bruhat-Tits building associated with SL(n,Q_p).
Cartan limits of SL(n, Q_p)read_more
HG G 43
25 October 2017
15:45-16:45 		C.S. Aravinda
Tata Institute of Fundamental Research, Bangalore
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Geometry Seminar

Title Geodesic conjugacies in nonpositive curvature
Speaker, Affiliation C.S. Aravinda, Tata Institute of Fundamental Research, Bangalore
Date, Time 25 October 2017, 15:45-16:45
Location HG G 43
Abstract The question of whether a time-preserving geodesic conjugacy determines a closed, negatively curved Riemannian manifold up to anisometry is one of the central problems in Riemannian geometry. While an answer to the question in this generality has yet remained elusive, this talk will briefly give an overview and discuss a minor improvement of a known result.
Geodesic conjugacies in nonpositive curvatureread_more
HG G 43
1 November 2017
15:45-16:45 		Raphael Zentner
University of Regensburg
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Geometry Seminar

Title Irreducible SL(2,C)-representations of integer homology 3-spheres
Speaker, Affiliation Raphael Zentner, University of Regensburg
Date, Time 1 November 2017, 15:45-16:45
Location HG G 43
Abstract We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).
Irreducible SL(2,C)-representations of integer homology 3-spheresread_more
HG G 43
8 November 2017
15:45-16:45 		Jonas Beyrer
Universität Zürich
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Geometry Seminar

Title Cross ratios on Furstenberg boundaries
Speaker, Affiliation Jonas Beyrer, Universität Zürich
Date, Time 8 November 2017, 15:45-16:45
Location HG G 43
Abstract A rank one symmetric space of non-compact type carries naturally a cross ratio on its visual boundary, which has many interesting applications. In particular the cross ratio characterizes the isometry group by its boundary action. We will use a similar geometric construction as for a rank one space to define cross ratios on Furstenberg boundaries of higher rank symmetric spaces of non-compact type. By showing several properties of those cross ratios, in particular that they characterize the isometry group of the symmetric space, we motivate that we get a reasonable generalization of the rank one case.
Cross ratios on Furstenberg boundariesread_more
HG G 43
15 November 2017
15:45-16:45 		Dr. Nicolas Matte Bon
ETH Zurich, Switzerland
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Geometry Seminar

Title Actions and homomorphisms of topological full groups
Speaker, Affiliation Dr. Nicolas Matte Bon, ETH Zurich, Switzerland
Date, Time 15 November 2017, 15:45-16:45
Location HG G 43
Abstract To any group or pseudogroup of homeomorphisms of the Cantor set one can associate a larger (countable) group, called the topological full group. It is a complete invariant of the groupoid of germs of the underlying action (every isomorphism between full groups is implemented by a conjugacy of the corresponding pseudogroups). First I'll discuss a result relating the growth of the orbits of a pseudogroup to a combinatorial fixed point property of its full group, and explain an application related to co-amenability and growth of Schreier graphs of finitely generated groups. Next, I will discuss a theorem on the possible actions on topological full groups on compact spaces, and apply it to show that arbitrary homomorphisms between full groups are often implemented at the level of the groupoids of germs. These results are proven working in the Chabauty space.
Actions and homomorphisms of topological full groupsread_more
HG G 43
22 November 2017
15:45-16:45 		Rémi Coulon
Université de Rennes 1 - CNRS
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Geometry Seminar

Title Growth gap in hyperbolic groups and amenability (joint work with Françoise Dal'Bo and Andrea Sambusetti)
Speaker, Affiliation Rémi Coulon, Université de Rennes 1 - CNRS
Date, Time 22 November 2017, 15:45-16:45
Location HG G 43
Abstract Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growthrate is bounded above by the one of G. In this work we are interested int he following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuck and Cohenproved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuck and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory. We focus here one the class of Gromov hyperbolic groups and propose a framework that encompasses both the combinatorial and the geometric point of view. More precisely, we prove that if G is a hyperbolic group acting properly co-compactly on a metric space X which is either a Cayley graph of G or a CAT(-1) space, then the growth rate of H and G coincide if and only if H is co-amenable in G. In addition, if G has Kazhdan property (T), we prove that there is a gap between the growth rate of G and the one of its infinite index subgroups.
Growth gap in hyperbolic groups and amenability (joint work with Françoise Dal'Bo and Andrea Sambusetti)read_more
HG G 43
29 November 2017
15:45-16:45 		Daniel Woodhouse
Technion – Israel Institute of Technology
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Geometry Seminar

Title Determining Commensurability of Simple Surface Amalgams Via a Common Model Geometry
Speaker, Affiliation Daniel Woodhouse, Technion – Israel Institute of Technology
Date, Time 29 November 2017, 15:45-16:45
Location HG G 43
Abstract A model geometry for a finitely generated group is a proper geodesic metric space on which the group acts properly and cocompactly. If two groups have a common model geometry, the Milnor-Schwarz Lemma tells us that the groups are quasiisometric. In contrast, two quasi-isometric groups do not, in general, have a common model geometry. A simple surface amalgam is obtained by taking a finite collection of compact surfaces, each with a single boundary component, and gluing them together by identifying their boundary curves. We consider the fundamental groups of such spaces and show that commensurability is determined by having a common model geometry. This gives a relatively simple family of groups that are quasi-isometric, but are neither commensurable, nor act on the same common model geometry.
Determining Commensurability of Simple Surface Amalgams Via a Common Model Geometryread_more
HG G 43
6 December 2017
15:45-16:45 		Alain Valette
Université de Neuchâtel
Details

Geometry Seminar

Title Diameters in box spaces
Speaker, Affiliation Alain Valette, Université de Neuchâtel
Date, Time 6 December 2017, 15:45-16:45
Location HG G 43
Abstract If $G$ is a finitely generated residually finite group, and $(N_i)_{i>0}$ is a decreasing sequence of finite index normal subgroups with trivial intersection, we study the diameter of $G/N_i$ ​as a function of the order $|G/N_i|$. For $0<\alpha\geq 1$, we say (after Breuillard and Tointon) that the box space $\square_(N_i)G$ has property $D_\alpha$, if the diameter of $G/N_i$ grows at least like $|G/N_i|^\alpha$. We show that a box space has property $D_1$ if and only if G is virtually cyclic; and that a group G admits some box space with property $D_\alpha$ (for some $\alpha>0$) if and only if G virtually maps onto $\mathbb{Z}$. For the lamplighter group and for a lattice in SOL, we provide explicit examples of box spaces with and without property $D_\alpha$.
Diameters in box spacesread_more
HG G 43
13 December 2017
15:45-16:45 		Dawid Kielak
Universität Bielefeld
Details

Geometry Seminar

Title The smallest quotients of Aut(F_n) and mapping class groups
Speaker, Affiliation Dawid Kielak, Universität Bielefeld
Date, Time 13 December 2017, 15:45-16:45
Location HG G 43
Abstract We will see how the classification of finite simple groups and some representation theory can be used to determine the smallest quotients of Aut(F_n) and mapping class groups. In both cases the smallest non-abelian quotients are obtained by first passing to the automorphism of the abelianisation of, respectively, free and surface groups, and then taking the smallest quotient of the relevant algebraic group.
The smallest quotients of Aut(F_n) and mapping class groupsread_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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