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Gromov conjectured that any irreducible lattice in a symmetric space of rank at least [Formula: see text] should have at most polynomial Dehn function. We prove that the lattice [Formula: see text] has quadratic Dehn function when [Formula: see text]. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

A discrete group with word-length (G, L) is [Formula: see text]-isocohomological for a bounding classes [Formula: see text] if the comparison map from [Formula: see text]-bounded cohomology to ordinary cohomology (with coefficients in ℂ) is an isomorphism; it is strongly [Formula: see text]-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong [Formula: see text]-isocohomologicality. In particular, we show strong [Formula: see text]-isocohomologicality for an FP∞ group G if all of the weighted G-sensitive Dehn functions are [Formula: see text]-bounded. Such groups include all [Formula: see text]-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of [Formula: see text]-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relativeFP∞ property for groups with respect to a collection of subgroups. Applications for computing [Formula: see text]-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

We extend the notion of geometric intersection numbers 0 and 1, for circles in the Dehn quandle, to general quandles, assuming existence of elements having certain algebraic properties in the latter. These properties enable the construction of many useful 2 and 3 cycles in Dehn quandle homology. Particularly, we construct a special 2-cycle homology representative and an operation which promotes it to higher dimensional cycles of the same type, in the Dehn quandle, and show how to do so in any quandles admitting such elements. For such quandles, this gives some non-triviality results in the quandle homology. We look at a secondary rack formed by tuples of elements of the Dehn quandle and using the ideas above, obtain some basic non-triviality results in its homology as well. We also give some initial possibilities for its application in representing monodromy of singularities of elliptic fibrations.

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

It is well-known that the "all right" metric structure on the Dehn complex of a knot is non-positively curved, or CAT(0), if and only if the knot projection is prime and alternating. We extend these ideas to spatial graphs and introduce the idea of ψ-alternating graphs as the appropriate analog of alternating knots.

Let K be a knot in S3 and suppose that K(r) is a reducible manifold. Howie has proved that the number of connected summands is at most 3 [10, Corollary 5.3]. Furthermore, he showed that if K(r) is the connected sum of exactly three irreducible 3-manifolds, then K(r) = L(p1, q1)♯L(p2, q2)♯M, where L(p1, q1) and L(p2, q2) are a lens spaces and M is a ℤ-homology sphere. In this paper we study this specific case and we show that |r| ≤ (b-2)(b-1), for any knot with bridge number b.

We construct a finitely presented group G with non-quadratic Dehn function f majorizable by a quadratic function on arbitrary long intervals.

We give an example of a finitely presented group G with two non-π1-equivalent asymptotic cones.

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection [Formula: see text] as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

Let M be a compact, orientable, irreducible 3-manifold with an incompressible torus boundary T and γ a longitudinal slope on T, which bounds a surface F of genus 2. Suppose there exists a slope r that produces an essential 2-sphere S by Dehn filling. Let q be the minimal geometric intersection number between the essential 2-sphere and the core of the Dehn filling. Matignon and Sayari [5] proved that either q = 2 or the minimal geometric intersection number between γ and r is bounded by 3. Here, we construct an example of a hyperbolic 3-manifold realizing that bound.

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful representation in the Cuntz C*-algebra. For the finitely presented simple group T_fin (also known as V) we show that the word-length and the table size satisfy an n log n relation, just like the symmetric groups. We show that the word problem of T_fin belongs to the parallel complexity class AC^1 (a subclass of plynomial time). We show that the generalized word problem of T_fin is undecidable. We study the distortion functions of T_fin and we show that T_fin contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of T_fin, the set of all Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

We link the Dehn function of finitely presented groups to the length-of-proofs function in propositional proof complexity.

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

In the present paper, we will study the creation of projective planes by Dehn surgery on knots in the 3-sphere. It is shown that a projective plane cannot be created by Dehn surgery on a crosscap number two knot. As a corollary, we will prove that crosscap number two knots satisfy the projective space conjecture, which asserts that the projective 3-space cannot be obtained by Dehn surgery on a non-trivial knot in the 3-sphere.

We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyone those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang., Comment: Ammended version with corrections to original and additional appendix on Alexander polynomials

We study the Dehn functions of amalgamations, introducing the notion of strongly undistorted subgroups. Using this, we give conditions under which taking an amalgamation does not increase the Dehn function, generalizing one aspect of the combination theorem of Bestvina and Feighn. To obtain examples of strongly undistorted subgroups, we define and study the relative Dehn function of pairs of groups. As a result we obtain a new method of constructing examples of pairs of groups that are relatively hyperbolic in the sense of Farb.

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

A virtual link comes with a variety of link complements. This paper is concerned with the Dehn space, a pseudo-manifold with boundary, and the Dehn complex, a two-dimensional spine of the Dehn space. In the classical case where the link is planar, the Dehn space is the link complement in the 3-sphere. We study topological and geometric properties of the Dehn complex of a virtual link. Among other things, we show that every finitely presented group is the fundamental group of a Dehn complex, and that the Dehn complex of any alternating triple of an alternating virtual link is a non-positively curved squared complex.