Related papers: Statistics and compression of scl

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $\Gamma$ which…

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric…

Consider a closed surface $S$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $S$ with finite first moment with respect to some hyperbolic metric on $S$. Corresponding to each point in…

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property…

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on $\Z^d$, where the transition probability from $x$ to $y$ is given by $w_{t, x,y}|x-y|^{-d-\alpha}$ with…

A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the…

We investigate the properties of word lengths of elements from a three-reflection symmetric generating set of the dihedral group $D_n$. Specifically, we provide the upper bound $\lambda_1(D_n,S) \leq \lfloor\frac{n}{2}\rfloor + 1$ for a…

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of…

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes…

We investigate translation length functions for two-generated groups acting by isometries on $\Lambda$-trees, where $\Lambda$ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of…

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage and last-exit times. In our…

We show that for any positive integer $n$ there exists a constant $C(n)>0$ such that any $n$-generated group $G$, which acts by isometries on a $\delta$-hyperbolic space (with $\delta>0$), is either free or has a nontrivial element with…

For finitely supported random walks on finitely generated groups $G$ we prove that the identity map on $G$ extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being…

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and…

Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating…

We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental…

We give an upper and lower estimate for the maximal commutator length of a noncentral element of the elementary subgroup of the general linear group over a skew-field based on the maximal commutator length of an element of the…

Consider a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n goes to infinity, these permutations viewed as operators on…

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…

We prove that quantum information propagates with a finite velocity in any model of interacting bosons whose (possibly time-dependent) Hamiltonian contains spatially local single-boson hopping terms along with arbitrary local…