Giulio Tiozzo
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Giulio Tiozzo has a total of 37 co-authors. Here are the top co-authors by number of shared publications:
- Carlo Carminati
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
4 shared publications - Joseph Maher
CUNY College of Staten Island and CUNY Graduate Center, Staten Island, USA
4 shared publications - Samuel J. Taylor
Temple University, Philadelphia, USA
4 shared publications - Ilya Gekhtman
Department of Mathematics, Technion—Israel Institute of Technology, Technion City, Israel
3 shared publications - Adam Epstein
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
2 shared publications - Anna Zdunik
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
2 shared publications
Latest Publications
- Open Access
Random walks on cocompact Fuchsian and Kleinian groups
The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $Γ$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups.Open AccessRandom walks on cocompact Fuchsian and Kleinian groups
The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $Γ$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups. - Open Access
Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
We consider symmetric random walks on discrete, Zariski-dense subgroups $Γ$ of a semisimple Lie group $G$ with Property (T). We prove that if $Γ$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank.Open AccessSingularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
We consider symmetric random walks on discrete, Zariski-dense subgroups $Γ$ of a semisimple Lie group $G$ with Property (T). We prove that if $Γ$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank. - 29 July 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Groups, Geometry, and Dynamics
Genericity of contracting geodesics in groups
Let G be a finitely generated group and \operatorname{Cay}(G, S) be the Cayley graph of G with respect to a finite generating set S . We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in \operatorname{Cay}(G, S) .
- 29 July 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Groups, Geometry, and Dynamics
Genericity of contracting geodesics in groups
Let G be a finitely generated group and \operatorname{Cay}(G, S) be the Cayley graph of G with respect to a finite generating set S . We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in \operatorname{Cay}(G, S) .
- 5 March 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (5) , 3297-3348
Master teapots and entropy algorithms for the Mandelbrot set
We construct an analogue of W. Thurston’s “Master teapot” for each principal vein in the Mandelbrot set, and generalize geometric properties known for the corresponding object for real maps. In particular, we show that eigenvalues outside the unit circle move continuously, while we show “persistence” for roots inside the unit circle. As an application, this shows that the outside part of the corresponding “Thurston set” is path connected. In order to do this, we define a version of kneading theory for principal veins, and we prove the equivalence of several algorithms that compute the core entropy.
- 5 March 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (5) , 3297-3348
Master teapots and entropy algorithms for the Mandelbrot set
We construct an analogue of W. Thurston’s “Master teapot” for each principal vein in the Mandelbrot set, and generalize geometric properties known for the corresponding object for real maps. In particular, we show that eigenvalues outside the unit circle move continuously, while we show “persistence” for roots inside the unit circle. As an application, this shows that the outside part of the corresponding “Thurston set” is path connected. In order to do this, we define a version of kneading theory for principal veins, and we prove the equivalence of several algorithms that compute the core entropy.
- Open Access
A cell decomposition for marked cycle curves
We describe a family $\textrm{Cyc}_p(\mathcal{F})$ of marked cycle curves that parameterize the cycles of period $p$ of a given family $\mathcal{F}$ of dynamical systems. We produce algorithms to compute a canonical cell decomposition for the marked cycle curves over the family $\textrm{Per}_1(0)$ of quadratic polynomials as well as over the family $\textrm{Per}_2(0)$ of quadratic rational maps with a critical 2-cycle. We obtain formulas for the number of $d$-cells in these decompositions, giving rise to e.g. a formula for their genus.Open AccessA cell decomposition for marked cycle curves
We describe a family $\textrm{Cyc}_p(\mathcal{F})$ of marked cycle curves that parameterize the cycles of period $p$ of a given family $\mathcal{F}$ of dynamical systems. We produce algorithms to compute a canonical cell decomposition for the marked cycle curves over the family $\textrm{Per}_1(0)$ of quadratic polynomials as well as over the family $\textrm{Per}_2(0)$ of quadratic rational maps with a critical 2-cycle. We obtain formulas for the number of $d$-cells in these decompositions, giving rise to e.g. a formula for their genus. - Top Cited PapersOpen Access
- 18 July 2024
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 28 (4) , 1829-1889
Sublinearly Morse boundary, II: Proper geodesic spaces
We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space X and any sublinear function Ä, we construct a boundary for X , denoted by @ Ä X , that is quasi-isometrically invariant and metrizable. As an application, we show that when G is the mapping class group of a finite type surface or a relatively hyperbolic group, with minimal assumptions, the Poisson boundary of G can be realized on the Ä-Morse boundary of G equipped with the word metric associated to any finite generating set. 20F65, 20F67, 57M07
- 18 July 2024
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 28 (4) , 1829-1889
Top Cited PapersOpen AccessSublinearly Morse boundary, II: Proper geodesic spaces
We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space X and any sublinear function Ä, we construct a boundary for X , denoted by @ Ä X , that is quasi-isometrically invariant and metrizable. As an application, we show that when G is the mapping class group of a finite type surface or a relatively hyperbolic group, with minimal assumptions, the Poisson boundary of G can be realized on the Ä-Morse boundary of G equipped with the word metric associated to any finite generating set. 20F65, 20F67, 57M07
- Open Access
Genericity of contracting geodesics in groups
Let G be a finitely generated group and Cay(G, S) be the Cayley graph of G with respect to a finite generating set S. We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in Cay(G, S).
Open AccessGenericity of contracting geodesics in groups
Let G be a finitely generated group and Cay(G, S) be the Cayley graph of G with respect to a finite generating set S. We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in Cay(G, S).
- 1 January 2023
- Published by Indiana University Mathematics Journal in Indiana University Mathematics Journal
- Vol. 72 (1) , 301-329
A central limit theorem for the degree of a random product of Cremona transformations
- 1 January 2023
- Published by Indiana University Mathematics Journal in Indiana University Mathematics Journal
- Vol. 72 (1) , 301-329
A central limit theorem for the degree of a random product of Cremona transformations
- Open Access
- 21 November 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10, e102
The fundamental inequality for cocompact Fuchsian groups
We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.
- 21 November 2022
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 10, e102
Open AccessThe fundamental inequality for cocompact Fuchsian groups
We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.
- Open Access
- 1 October 2022
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 158 (10) , 1980-2013
Central limit theorems for counting measures in coarse negative curvature
We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$ . Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.- 1 October 2022
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 158 (10) , 1980-2013
Open AccessCentral limit theorems for counting measures in coarse negative curvature
We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$ . Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.