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We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichm\"{u}ller metric., Comment: 22 pages, no figures, exposition improved and reference updated

We prove that the mapping class group of a Heegaard splitting with a distance of at least 3 is finite. However, we have constructed a counterexample with a distance of 2 that disproves this assertion. In addition, the fact that the mapping class group of a Heegaard splitting with a distance of at most 1 is infinite, when combined with our results, provides an answer to the question of the finiteness of mapping class groups as viewed from Heegaard distance., Comment: 7 Pages. Comments welcome!

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $ K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $ t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$. After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance Heegaard splitting or $p/q$ is far away from a fixed subset in the Farey graph, then $t(K^{\star})= t(K)+1$. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

We prove that for any distance at least 3 Heegaard splitting and a boundary component $F$, there is a diameter finite ball in the curve complex $\mathcal {C}(F)$ so that it contains all distance degenerate curves or slopes in $F$., Comment: 8figures

From the view of Heegaard splitting, it is known that if a closed orientable 3-manifold admits a distance at least three Heegaard splitting, then it is hyperbolic. However, for a closed orientable 3-manifold admitting only distance at most two Heegaard splittings, there are examples shows that it could be reducible, Seifert, toroidal or hyperbolic. According to Thurston's Geometrization conjecture, the most important piece of eight geometries is hyperbolic. Thus to read out a hyperbolic 3-manifold from a distance two Heegaard splittings is critical in studying Heegaard splittings. Inspired by the construction of hyperbolic 3-manifolds with a distance two Heegaard splitting [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1, 231-255], we introduce the definition of a locally large geodesic in curve complex and furthermore the locally large distance two Heegaard splitting. Then we prove that if a 3-manifold admits a locally large distance two Heegaard splitting, then it is a hyperbolic manifold or an amalgamation of a hyperbolic manifold and a seifert manifold along an incompressible torus, i.e., almost hyperbolic, while the example in Section 3 shows that there is a non hyperbolic 3-manifold in this case. After examining those non hyperbolic cases, we give a sufficient and necessary condition for a hyperbolic 3-manifold when it admits a locally large distance two Heegaard splitting., Comment: title changed; one picture added; one picture changed; the theorem 1.4 is updated into a new full version

In this paper, we prove that (1) For any integers $n\geq 1$ and $g\geq 2$, there is a closed 3-manifold $M_{g}^{n}$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g, n)$ is $(2, 1)$. Furthermore, $M_{g}^{n}$ can be chosen to be hyperbolic except that the pair of $(g, n)$ is $(3, 1)$. (2) For any integers $g\geq 2$ and $n\geq 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$., Comment: 20 pages, 8 figures 2 References Added, title changed