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Tatiana Nagnibeda

Publications and source records attributed to Tatiana Nagnibeda.

33 records · Page 2Linked to original sources
arXiv2016

Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution

There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.

math.GR↗
arXiv2016

Weakly maximal subgroups in regular branch groups

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic.

math.GR↗
arXiv2016

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Every self-similar group acts on the space $X^ω$ of infinite words over some alphabet $X$. We study the Schreier graphs $Γ_w$ for $w\in X^ω$ of the action of self-similar groups generated by bounded automata on the space $X^ω$. Using sofic subshifts we determine the number of ends for every Schreier graph $Γ_w$. Almost all Schreier graphs $Γ_w$ with respect to the uniform measure on $X^ω$ have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.

math.GR↗
arXiv2015

Small spectral radius and percolation constants on non-amenable Cayley graphs

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $Γ$, does there exist a generating set $S$ such that the Cayley graph $(Γ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(Γ,S)<p_u(Γ,S)$? We show that this is true if $Γ$ contains an infinite normal subgroup $N$ such that $Γ/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $Γ$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to free Burnside groups $B(n,p)$ with $n \geq 2, p \geq 665$. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group.

math.GR↗
arXiv2014

On transitivity and (non)amenability of Aut(F_n) actions on group presentations

For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\geq \operatorname{rank}(G)$, describes the action of the group $\operatorname{Aut}F_n$ of automorphisms of the free group $F_n$ on generating $n$-tuples of G by elementary Nielsen moves. The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\operatorname{Aut}F_n$, $n\geq 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max\{2,\operatorname{rank}(G)\}$ when $G$ is indicable, and for $n$ big enough when $G$ is elementary amenable. We give an explicit description of $N_d(G)$ for relatively free (in some variety) groups of rank $d$ and discuss their connectedness and nonamenability. Examples considered include free polynilpotent groups and free Burnside groups.

math.GR↗
arXiv2014

Weighted Coxeter graphs and generalized geometric representations of Coxeter Groups

We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.

math.CO↗
arXiv2012

Subset currents on free groups

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.

math.GR↗
arXiv2012

Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups

The main result of this paper is a rigorous proof of criticality and an explicit computation of critical exponents for the decay of avalanches in the Abelian sandpile model (ASM) on a large family of infinite graphs. We begin by introducing the notion of criticality of the ASM for limits of finite graphs in local convergence, which naturally leads to the question about criticality of the ASM in the random weak limit. Our main technical ingredient is a sufficient condition for almost sure criticality of the ASM on sequences of finite cacti (i.e., separable graphs whose blocks are cycles or single edges) under the assumption that the random weak limit is almost surely 1-ended. Examples that allow explicit computations of the critical exponents come from actions of finitely generated groups on regular rooted trees, by automorphisms. Restricting the action to the consecutive levels of the tree defines a sequence of finite graphs whose limits in the local convergence are orbital Schreier graphs for the action of the group on the boundary of the tree. In the case of iterated monodromy groups of complex polynomials, these graphs are cacti, and we show that for 1-ended ones, the critical exponent for the decay of the mass of avalanches depends on the growth of the graph. The well-known Basilica group related to $z^2-1$ gives rise to uncountably many 4-regular one-ended graphs of quadratic growth with the critical exponent for the mass of avalanches equal to 1; as well as uncountably many new non-critical examples of quadratic growth (thus not quasi-isometric to $\mathbb{Z}$). Another iterated monodromy group that we consider provides uncountably many graphs with the critical exponent equal to $2\log 2/\log 3 >1$. Finally, we also exhibit graphs of polynomial growth with arbitrarily small critical exponent.

math.PR↗
arXiv2011

Counting dimer coverings on self-similar Schreier graphs

We study partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions of finitely generated groups by automorphisms on rooted trees, and thus their edges are naturally labeled by the generators of the group. It is thus natural to consider weight functions on these graphs taking different values according to the labeling. We study in detail the well-known example of the Hanoi Towers group $H^{(3)}$, closely related to the Sierpiński gasket.

math.CO↗
arXiv2011

Ergodic properties of boundary actions and Nielsen--Schreier theory

We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup $H$ of a free group $F$ on the boundary $\partial F$ with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen--Schreier generators in $H$ associated with a geodesic spanning tree in the Schreier graph $X=H\backslash F$. We give several (mod 0) equivalent descriptions of the Hopf decomposition of the boundary into the conservative and the dissipative parts. Further we relate conservativity and dissipativity of the action with the growth of the Schreier graph $X$ and of the subgroup $H$ ($\equiv$ cogrowth of $X$), respectively. We also construct numerous examples illustrating connections between various relevant notions.

math.GR↗
arXiv2010

Geometric entropy of geodesic currents on free groups

A \emph{geodesic current} on a free group $F$ is an $F$-invariant measure on the set $\partial^2 F$ of pairs of distinct points of $\partial F$. The space of geodesic currents on $F$ is a natural companion of Culler-Vogtmann's Outer space $cv(F)$ and studying them together yields new information about both spaces as well as about the group $Out(F)$. The main aim of this paper is to introduce and study the notion of {\it geometric entropy} $h_T(μ)$ of a geodesic current $μ$ with respect to a point $T$ of $cv(F)$, which can be viewed as a length function on $F$. The geometric entropy is defined as the slowest rate of exponential decay of $μ$-measures of bi-infinite cylinders in $F$, as the $T$-length of the word defining such a cylinder goes to infinity. We obtain an explicit formula for $h_{T'}(μ_T)$, where $T,T'$ are arbitrary points in $cv(F)$ and where $μ_T$ denotes a Patterson-Sullivan current corresponding to $T$. It involves the volume entropy $h(T)$ and the extremal distortion of distances in $T$ with respect to distances in $T'$. It follows that, given $T$ in the projectivized outer space $CV(F)$, $h_{T'}(μ_T)$ as function of $T'\in CV(F)$ achieves a strict global maximum at $T'=T$. We also show that for any $T\in cv(F)$ and any geodesic current $μ$ on $F$, $h_T(μ)\le h(T)$, where the equality is realized when $μ=μ_T$. For points $T\in cv(F)$ with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measure-theoretic entropy.

math.GR↗
arXiv2010

Schreier graphs of the Basilica group

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{Γ_n\}_{n\geq 1}$, where $Γ_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $Γ_ξ$, $ξ\in \partial T$. Denoting by $ξ_n$ the prefix of length $n$ of the infinite ray $ξ$, the rooted graph $(Γ_ξ,ξ)$ is then the limit of the sequence of finite rooted graphs $\{(Γ_n,ξ_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(Γ_ξ,ξ)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $ξ$.

math.GR↗
arXiv2010

Partition functions of the Ising model on some self-similar Schreier graphs

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk's group of intermediate growth; the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group; and the Hanoi Towers group $H^{(3)}$ closely related to the Sierpinsky gasket.

math.CO↗
arXiv2006

The Patterson-Sullivan embedding and minimal volume entropy for outer space

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $k\ge 2$ the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank $k$ and without degree-one vertices is equal to $(3k-3)\log 2$ and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs.

math.GR↗