We study the topological dynamics of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. We show that, for any suitable probability measure $\mu $ , random walks with respect to $\mu $ will produce elements with strong mixing properties for this action asymptotically almost surely. In particular, when the group has no finite normal subgroups, this implies that the action is highly topologically transitive. Along the way, we prove technical results about convex cocompact subgroups that allow us to extend some results on random walks of Abbott and the first author.

Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a persistence diagram. To utilize machine and deep learning methods on persistence diagrams, These diagrams are further summarized by transforming them into functions. In this paper we investigate the stability and injectivity of a class of smooth, one-dimensional functional summaries called Gaussian persistence curves.

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Agol–Liu (J. Am. Math. Soc. 25(1):151–187, 2012) and Reid–Wang–Zhou (Acta Math. Sin. Engl. Ser. 18(1):157–172, 2002).

We show that if a right-angled Artin group [Formula: see text] has a non-trivial, minimal action on a tree [Formula: see text] with more than two ends, then [Formula: see text] contains a separating subgraph [Formula: see text] such that [Formula: see text] stabilizes an edge in [Formula: see text].

We prove the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups. In real dimension three, where a result of Hempel ensures that the fundamental group is always residually finite, we then provide a Price type inequality proof of a well-known result of Lott and Lueck. Finally, we give several classes of higher graph manifolds which do indeed have residually finite fundamental groups.

It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $\langle H, R\rangle\cong H\ast R$ with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in $G$. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when $G$ is the mapping class group of a surface and $H$ is a convex cocompact subgroup we show that $\langle H, R\rangle$ is convex cocompact and isomorphic to $ H\ast R$.

In topological data analysis, persistent homology is used to study the "shape of data".Persistent homology computations are completely characterized by a set of intervals called a bar code.It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise".We give evidence to dispute this thesis, showing that the short intervals encode geometric information.Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled.We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space.In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.

Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. Persistent homology is one of the most commonly used tools in TDA, in part because it can be easily visualized in the form of a persistence diagram. However, performing machine learning algorithms directly on persistence diagrams is a challenging task, and so a number of summaries have been proposed which transform persistence diagrams into vectors or functions. Many of these summaries fall into the persistence curve framework developed by Chung and Lawson. We extend this framework and introduce new class of smooth persistence curves which we call Gaussian persistence curves. We investigate the statistical properties of Gaussian persistence curves and apply them to texture datasets: UIUCTex and KTH. Our classification results on these texture datasets perform competitively with the current state-of-arts methods in TDA.

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an appropriate sense. As an application, we prove that any relatively hyperbolic group with equationally noetherian peripheral subgroups is itself equationally noetherian.

We show that if a finitely generated group$G$has a nonelementary WPD action on a hyperbolic metric space$X$, then the number of$G$-conjugacy classes of$X$-loxodromic elements of$G$coming from a ball of radius$R$in the Cayley graph of$G$grows exponentially in$R$. As an application we prove that for$N\geq 3$the number of distinct$\text{Out}(F_{N})$-conjugacy classes of fully irreducible elements$\unicode[STIX]{x1D719}$from an$R$-ball in the Cayley graph of$\text{Out}(F_{N})$with$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$of the order of$R$grows exponentially in$R$.

We characterize when (and how) a Right-Angled Artin group splits nontrivially over an abelian subgroup.

Higher education institutions are often concerned about retention rates, particularly among underprepared students. This study examines the effects of Counselors providing Resources, Integration, Skill Development, and Psychosocial Support (CRISP), which is a low-cost counseling model focused on increasing the academic success and retention of underprepared undergraduates (n = 273). Results indicated participation in CRISP was significantly related to higher first-year grade point average and retention to the second and third years of college.

We generalize a version of small cancellation theory to the class of acylindrically hyperbolic groups. This class contains many groups which admit some natural action on a hyperbolic space, including non-elementary hyperbolic and relatively hyperbolic groups, mapping class groups, and groups of outer automorphisms of free groups. Several applications of this small cancellation theory are given, including to Frattini subgroups and Kazhdan constants, the construction of various "exotic" quotients, and to approximating acylindrically hyperbolic groups in the topology of marked group presentations.