Organizers: Sahana Hassan Balasubramanya and Thomas Ng
Given a group $G$ acting cocompactly on a suitable simply connected cell complex $X$ with relatively hyperbolic cell stabilizers, we show $G$ itself is relatively hyperbolic. Building on work of Dahmani and Martin, the proof constructs a model for the Bowditch boundary by gluing together the boundaries of cell stabilizers. More generally, any cocompact action on a cell complex $X$ induces an algebraic \emph{complex of groups} decomposition which generalizes Bass--Serre theory in the case where $X$ is 1--dimensional. This model connects this algebraic decomposition with a topological decomposition of the boundary which we hope will be useful for answering other questions.
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Let $G$ be a group acting by isometries on a hyperbolic space $X$. Given geometrically natural subgroups $H$ and $K$ of $G$, it is natural to ask whether $ \langle H, K \rangle $ inherits the geometric properties of $H$ and $K$, and whether $ \langle H, K \rangle $ admits a nice algebraic structure. In a classical work of Gitik we receive an answer in the case where $G$ is a hyperbolic group, and $H$ and $K$ are quasiconvex. In more recent work of Martínez-Pedroza and Sisto, we receive an answer in the case where $G$ is relatively hyperbolic, and $H$ and $K$ are relatively quasiconvex. In this talk, I will discuss a generalization of a combination theorem of Abbot and Manning that covers a broader class of geometrically natural subgroups of such a group $G$. This is still a work in progress.
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Brian Udall proved that there is a short exact sequence of the form: $$1 \xrightarrow{} Twist(M_{\Gamma})\xrightarrow{} Map(M_{\Gamma})\xrightarrow{\Psi} Map(\Gamma) \xrightarrow{}1,$$ where $Twist(M_{\Gamma})$ is the subgroup of $Map(M_{\Gamma})$ generated by sphere twists over sphere systems of $M_{\Gamma}.$ Udall also proved that this short exact sequence splits topologically. The purpose of this talk is to present an explicit formula for a section s of this short exact sequence.
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We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the graph has finitely many ends, we construct an explicit presentation for the so-called \emph{pure graph Houghton group} and investigate several of its algebraic and geometric properties. Additionally, we show that the graph Houghton groups are not commensurable with other known Houghton-type groups, namely the classical, surface, braided, handlebody, and doubled handlebody Houghton groups, demonstrating that this graph-based construction defines a genuinely new class of groups. This is joint work with Sanghoon Kwak, Brian Udall, and Jeremy West.
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A group is said to be bounded if it has finite diameter with respect to every bi-invariant metric. This is a strong rigidity property for large groups, limiting the large-scale geometry of the group and the types of geometric actions it can admit. Building on ideas of Burago, Ivanov, and Polterovich, Rybicki proved that the identity component of the homeomorphism group of a portable manifold is bounded. In this talk, I will present a simplified proof of this result by constructing a uniform normal generator for the group.
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The conjugator length function of a finitely generated group is the function f so that f(n) is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most n. This function provides a measure for the complexity of a direct approach to the Conjugacy Problem for the finitely generated group. I will discuss what functions can be realized as the conjugator length function of a finitely presented group and the connection of this function with other important invariants of finitely presented groups.
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The maximized hyperbolic space of a right-angled Coxeter group (RACG) can be obtained from its Davis complex by coning off all standard flats. This space serves as the top-level hyperbolic space in the hierarchically hyperbolic structure of the RACG, analogous to the curve graph for mapping class groups. We provide a necessary and sufficient condition on the defining graph under which the Gromov boundary of the maximized hyperbolic space is connected.
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In many useful settings, having a finite Bowen-Margulis-Sullivan (BMS) measure on a flow space allows people to normalize the BMS measure into a probability measure and facilitates powerful ergodic theoretic tools. This often leads to asymptotic estimates for counting orbital points and establishing equidistribution results. Hence, it is important to know when a dynamical system admits a finite BMS measure. In this talk, I will first introduce what is a BMS measure, and then state a criterion that detects the finiteness of BMS measure on a flow space associated to a discrete subgroup of higher rank semi-simple Lie group.
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The mapping class group of a locally finite graph Maps$(X)$ is the set of proper homotopy equivalences of $X$ up to proper homotopy. It is meant to be the analogue of the mapping class group of an infinite-type surface one dimension lower, but it also generalizes Out$(F_n)$ to a much larger class of possibly infinitely generated groups, establishing a "Big Out$(F_n)$." In this talk, I plan to define the mapping class group for a locally finite graph, discuss its topology, and give motivation. I will then discuss which locally finite graphs $X$ are such that Maps$(X)$ contains a dense conjugacy class. Along the way, we will discuss end spaces and their structures.
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The systole of a hyperbolic 3-manifold is the length of the shortest closed geodesic. Given a closed 3-manifold M and link L such that M\L is hyperbolic, the essential systole of M\L is the length of the shortest closed geodesic which is not nullhomotopic in M. In this talk, we will discuss and motivate the study of essential systoles of hyperbolic link complements, including their application towards answering a question of Freedman and Krushkal about the existence of "filling links" in closed 3-manifolds.
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A complex affine hyperplane arrangement is a locally finite collection of affine hyperplanes (complex codimension-1 subspaces) in a finite dimensional complex affine space. Since these subspaces have complex codimension 1, the complement of their union is a connected manifold. It is a broad, longstanding problem with many connections to different areas of mathematics to determine the arrangements for which this manifold is aspherical (has contractible universal cover). A subset of this problem dating back to the 1970s, commonly attributed to Arnol’d, Brieskorn, Thom, and Pham, concerns arrangements arising from reflection groups in real affine space. One approach that has seen success is to construct a cell complex which is homotopy equivalent to this complement and endow it with some kind of ("singular") non-positive curvature. Along these lines, by showing that a specific cell complex (based on a construction of Falk) carries an injective metric, we show that a broad class of affine arrangements (including the infinite families of affine reflection arrangements, modulo a conjecture about $D_n$-type) have aspherical complement. In particular, this provides some of the first examples of infinite affine arrangements which have aspherical complement, but do not arise from reflection groups. This is joint work with Jingyin Huang.
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Given a finite graph $\Gamma$, the associated *graph braid group* $B_n(\Gamma)$ is the fundamental group of the unordered $n$-point configuration space of $\Gamma$. Genevois classified which graph braid groups are Gromov hyperbolic and asked the question: When do these groups arise as $3$-manifold groups? In this talk, we give a partial answer for $B_3(\Theta_m)$ where $\Theta_m$ is the *generalized $\Theta$-graph*.
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Let G be a Polish group. A G-flow is a continuous action of G on a compact Hausdorff space. G is amenable if every G-flow has an invariant probability measure. G has metrizable universal minimal flow if every G-flow contains a metrizable G-flow. We consider the property of G having a metrizable Furstenberg boundary. This is a common weakening of amenability and having metrizable universal minimal flow. We prove a characterization of G having metrizable Furstenberg boundary in the spirit of Kechris-Pestov-Todorcevic, Bartosova, Moore, and Zucker. We show mapping class groups of infinite type surfaces never have a metrizable Furstenberg boundary. This strengthens a theorem of Long that such groups are not amenable. This is joint work with George Domat and Forte Shinko.
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I will discuss a new sufficient condition for when a braided link, a braid closure together with its braid axis, is bi-orderable meaning the fundamental group of its exterior admits an order invariant under both left and right multiplication. In 2006, Perron and Rolfsen provided a condition which ensures that an automorphism of a free group preserves a bi-order of the free group and shows that many fibered 3-manifolds are bi-orderable, including many exteriors of braided links. Recent work of Khanh Le and I provide another new criterion, via the Burau representation, for a free group automorphism to be order-preserving. Using the new criterion, we produce new examples of bi-orderable braided link groups including some examples produced from braids whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen. This work is partially supported by the NSF grant DMS-2213213.
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In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.
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Euclidean buildings (a.k.a. affine buildings and Bruhat-Tits buildings) are considered as a p-adic analogue of symmetric spaces. We show that there is no quasi-isometric embedding between the symmetric space of SL(n,R) and the Euclidean building of SL(n,Q_p). Generalizing this, we distinguish Ramanujan complexes constructed by Lubotzky-Samuels-Vishne as finite quotients of Euclidean buildings of PGL(n,F_p((y))) up to quasi-isometric embeddings. These complexes serve as high dimensional expanders with fruitful applications in mathematics and computer science.
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As a generalization of hyperbolic groups, the class of acylindrically hyperbolic groups includes many interesting examples and has has received considerable attention. In the world of hyperbolic groups, quasiconvex subgroups are important subjects. What would be a proper analog of quasiconvex subgroups in the context of acylindrically hyperbolic groups? In this talk, I will share my answer to this question and some more questions.
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Given a group *G* and a collection of codimension--one subgroups *H* of *G*, one can construct a CAT(0) cube complex on which *G* acts isometrically with no global fixed point. This is known as Sageev's construction. In this construction, codimension--one subgroups of *H* are commensurable with hyperplane stabilizers. By imposing certain conditions on *G* and *H*, one can promote the group action to a proper or cocompact one. A proper action is harder to obtain than a cocompact one. In this talk, we introduce a relative version of Groves--Manning's result on quasi-convexity in the Sageev construction. Let *(G,P)* be a finitely generated relatively hyperbolic group acting cocompactly and *P*-elliptically on a CAT(0) cube complex. In this setting, we show that vertex stabilizers are full relatively quasi-convex if and only if hyperplane stabilizers are full relatively quasi-convex.
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In the 1950s, Coxeter considered the quotients of braid groups given by adding the relation that all half Dehn twist generators have some fixed, finite order. He found a remarkable formula for the order of these groups in terms of some related Platonic solids. Despite the inspiring apparent connection between these objects, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. I'll explain recent work that gives an interpretation of the truncated 3-strand braid group that makes the connection with Platonic solids clear, using down-to-earth geometric and algebraic topological tools.
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In their paper, Branman, Domat, Hoganson and Lymann proved that if a topological group acts in a "nice" way on a simplicial graph, then the group has a well defined geometry that makes it quasi-isometric to the graph. These actions generalize a Svârc-Milnor action to the context of coarsely-boundedly (CB) generated Polish groups. We adapt these ideas to the context of locally bounded Polish groups and then construct an arc and curve model coarsely equivalent to Map(S) when Map(S) is locally bounded. We then use this model to show that the asymptotic dimension of Map(S) is infinite when S has a non-displaceable subsurface.
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