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Recent progress in cross-lingual relation and event extraction use graph convolutional networks (GCNs) with universal dependency parses to learn language-agnostic sentence representations such that models trained on one language can be applied to other languages. However, GCNs struggle to model words with long-range dependencies or are not directly connected in the dependency tree. To address these challenges, we propose to utilize the self-attention mechanism where we explicitly fuse structural information to learn the dependencies between words with different syntactic distances. We introduce GATE, a Graph Attention Transformer Encoder, and test its cross-lingual transferability on relation and event extraction tasks. We perform experiments on the ACE05 dataset that includes three typologically different languages: English, Chinese, and Arabic. The evaluation results show that GATE outperforms three recently proposed methods by a large margin. Our detailed analysis reveals that due to the reliance on syntactic dependencies, GATE produces robust representations that facilitate transfer across languages.